Short answer
Yes, and the Gospels show it. Early in his ministry, Jesus told a paralysed man that his sins were forgiven. He taught his followers to ask God (﷾) directly to forgive them. He said God (﷾) forgives those who forgive others. In his parable of the lost son, the father runs to meet his son and forgives him at once. No one is punished in the son's place. The prophets say the same. God (﷾) forgave the city of Nineveh when it turned from its sins. Through the prophet Hosea He said, "I desired mercy, and not sacrifice". God (﷾) is all-powerful and needs no blood to forgive. The Qur'an says Allah (﷾) forgives all sins to those who turn back to Him. It also says the blood of the animals sacrificed at the Hajj and at Eid does not reach Allah (﷾); only piety does.
Full explanation
Jesus forgave and taught forgiveness without blood
Jesus told a paralysed man, "Son, thy sins be forgiven thee", early in his ministry. He taught his disciples to pray, "forgive us our debts". He said at once what God's (﷾) forgiveness depends on: forgiving others. There is not a word of blood in the prayer or in its explanation. When a man asked him how to have eternal life, Jesus told him to keep the commandments.
“When Jesus saw their faith, he said unto the sick of the palsy, Son, thy sins be forgiven thee.”
“And forgive us our debts, as we forgive our debtors.”
“For if ye forgive men their trespasses, your heavenly Father will also forgive you: But if ye forgive not men their trespasses, neither will your Father forgive your trespasses.”
“And, behold, one came and said unto him, Good Master, what good thing shall I do, that I may have eternal life? And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.”
Jesus' own pictures of God's (﷾) forgiveness
In the parable of the lost son, the father sees the son a great way off, runs to him and kisses him. The son is forgiven when he comes home, and no one pays in his place. In another parable a tax collector stands far off and says only, "God be merciful to me a sinner". Jesus says he went home justified, made right with God (﷾).
“And he arose, and came to his father. But when he was yet a great way off, his father saw him, and had compassion, and ran, and fell on his neck, and kissed him.”
“And the publican, standing afar off, would not lift up so much as his eyes unto heaven, but smote upon his breast, saying, God be merciful to me a sinner. I tell you, this man went down to his house justified rather than the other: for every one that exalteth himself shall be abased; and he that humbleth himself shall be exalted.”
The prophets: turn back and live
Ezekiel says that if the wicked man turns from all his sins, none of them will be remembered against him. No sacrifice is recorded at Nineveh. Its people turned from their evil way, and God (﷾) spared them. Hosea sums up the principle: God (﷾) desires mercy and not sacrifice. Jesus quoted that verse twice.
“But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. All his transgressions that he hath committed, they shall not be mentioned unto him: in his righteousness that he hath done he shall live.”
“Again, when the wicked man turneth away from his wickedness that he hath committed, and doeth that which is lawful and right, he shall save his soul alive.”
“And God saw their works, that they turned from their evil way; and God repented of the evil, that he had said that he would do unto them; and he did it not.”
“For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.”
“But go ye and learn what that meaneth, I will have mercy, and not sacrifice: for I am not come to call the righteous, but sinners to repentance.”
“But if ye had known what this meaneth, I will have mercy, and not sacrifice, ye would not have condemned the guiltless.”
God (﷾) did not accept a substitute
After the golden calf, Moses offered to be blotted out of God's (﷾) book in place of Israel. God (﷾) did not take up the offer. Instead He gave a rule: whoever has sinned against Him is the one He blots out. The Qur'an states the same rule: no bearer of burdens bears another's burden.
“Yet now, if thou wilt forgive their sin—; and if not, blot me, I pray thee, out of thy book which thou hast written. And the Lord said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.”
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
“Nor can a bearer of burdens bear another's burdens if one heavily laden should call another to (bear) his load. Not the least portion of it can be carried (by the other). Even though he be nearly related. Thou canst but admonish such as fear their Lord unseen and establish regular Prayer. And whoever purifies himself does so for the benefit of his own soul; and the destination (of all) is to Allah.”
Is it just to punish the innocent?
No court would call it justice to punish an innocent man in place of the guilty, even a willing one. God's (﷾) own rule in Exodus is that the one who sinned answers for his sin. The Qur'an says no bearer of burdens bears another's. God (﷾) is not a court under another's law: His own word through Ezekiel says the one who turns from his sins shall live.
“Yet now, if thou wilt forgive their sin—; and if not, blot me, I pray thee, out of thy book which thou hast written. And the Lord said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.”
“Nor can a bearer of burdens bear another's burdens if one heavily laden should call another to (bear) his load. Not the least portion of it can be carried (by the other). Even though he be nearly related. Thou canst but admonish such as fear their Lord unseen and establish regular Prayer. And whoever purifies himself does so for the benefit of his own soul; and the destination (of all) is to Allah.”
“But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. All his transgressions that he hath committed, they shall not be mentioned unto him: in his righteousness that he hath done he shall live.”
An all-powerful God (﷾)
The Bible calls God the Almighty, and Jesus said that with God (﷾) all things are possible. A God who can do anything can forgive by His own will, as any person can forgive a debt. Christians reply that the question is God's (﷾) justice, not His power. But God's (﷾) own word through Ezekiel says that the wicked man who turns from all his sins shall live. None of them shall be mentioned to him. Forgiving the one who turns back is His justice and His mercy, by His own word.
“And when Abram was ninety years old and nine, the Lord appeared to Abram, and said unto him, I am the Almighty God; walk before me, and be thou perfect.”
“For with God nothing shall be impossible.”
“But Jesus beheld them, and said unto them, With men this is impossible; but with God all things are possible.”
“But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. All his transgressions that he hath committed, they shall not be mentioned unto him: in his righteousness that he hath done he shall live.”
“Again, when the wicked man turneth away from his wickedness that he hath committed, and doeth that which is lawful and right, he shall save his soul alive.”
What Christians believe
Christians hold that God's (﷾) justice requires sin to be paid for, and that only the blood of Jesus can pay. They quote Hebrews, which says that without shedding of blood there is no remission, meaning forgiveness. They quote Paul, a Christian missionary of the first century, who says believers are justified by his blood.
“And almost all things are by the law purged with blood; and without shedding of blood is no remission.”
“Much more then, being now justified by his blood, we shall be saved from wrath through him.”
What the Qur'an and the Prophet (ﷺ) teach
The Qur'an tells those who have wronged themselves not to despair of Allah's (﷾) mercy, for He forgives all sins. Whoever seeks His forgiveness will find Him forgiving. Adam asked forgiveness and God (﷾) turned to him. The Prophet (ﷺ) reported that Allah (﷾) says: whoever walks towards Me, I come to him running. Whoever meets Me with sins that fill the earth, but without associating anything with Me, I meet him with as much forgiveness.
“Say: "O my Servants who have transgressed against their souls! Despair not of the Mercy of Allah: for Allah forgives all sins: for He is Oft-Forgiving, Most Merciful.”
“If any one does evil or wrongs his own soul but afterwards seeks Allah's forgiveness, he will find Allah Oft-forgiving, Most Merciful.”
“And those who, having done something to be ashamed of, or wronged their own souls, earnestly bring Allah to mind, and ask for forgiveness for their sins,- and who can forgive sins except Allah?- and are never obstinate in persisting knowingly in (the wrong) they have done.”
“They said: "Our Lord! We have wronged our own souls: If thou forgive us not and bestow not upon us Thy Mercy, we shall certainly be lost."”
“Then learnt Adam from his Lord words of inspiration, and his Lord Turned towards him; for He is Oft-Returning, Most Merciful.”
Abu Dharr reports that the Prophet (ﷺ) said that Allah says: whoever brings a good deed will have ten like it or more; whoever draws a hand-span closer to Me, I draw an arm's length closer to him; whoever comes to Me walking, I come to him running; and whoever meets Me with sins that would fill the earth, but without associating anything with Me, I will meet him with forgiveness as great.
Christians say sin must be paid for, and only the blood of Jesus can pay: without shedding of blood there is no forgiveness.
Did Jesus forgive without blood?
Yes. Early in his ministry he told a paralysed man, “Son, thy sins be forgiven thee”.
“Forgiven in advance” Some say those sins were forgiven because of a payment still to come. The Bible gives another reason: Nineveh turned from its evil, and the tax collector in Jesus' parable humbled himself and asked for mercy.
What did he teach?
To ask God (﷾) directly for forgiveness. God (﷾) forgives those who forgive others. There is not a word of blood.
What did the prophets say?
When Nineveh turned from its evil, God (﷾) spared it, with no sacrifice recorded. Through Hosea, God (﷾) says He desires mercy, not sacrifice.
Did the Law itself allow it?
Yes. A man too poor for a lamb or birds could bring flour for his sin, and it was forgiven.
God (﷾) forgives whoever turns to Him, with no blood. Adam asked forgiveness and God (﷾) turned to him, and the Qur'an says Allah (﷾) forgives all sins to those who turn back to Him.
Islam has a sacrifice too: what the Qur'an says it does
Muslims sacrifice an animal at Eid al-Adha and in the Hajj, the pilgrimage to Mecca. The Qur'an says what reaches God (﷾): "It is not their meat nor their blood, that reaches Allah: it is your piety that reaches Him." It adds that He made the animals subject to people "that ye may glorify Allah for His Guidance to you". So in Islam, too, it is not blood that reaches God (﷾).
“It is not their meat nor their blood, that reaches Allah: it is your piety that reaches Him: He has thus made them subject to you, that ye may glorify Allah for His Guidance to you and proclaim the good news to all who do right.”
Verse by verse
Hebrews 9:22 — Epistle to the Hebrews, chapter 9, verse 22
“And almost all things are by the law purged with blood; and without shedding of blood is no remission.”
Cited for
Without the shedding of blood there is no forgiveness at all, so only the blood of Jesus can take sin away.
The response
The Law of Moses itself made an exception. It let a man too poor for a lamb or birds bring flour for his sin, "and it shall be forgiven him". Jesus forgave the paralysed man, and God (﷾) forgave Nineveh, with no blood at all.
Where it lands — The Bible itself records sins forgiven without blood.
Mark 2:5 — Gospel of Mark, chapter 2, verse 5
“When Jesus saw their faith, he said unto the sick of the palsy, Son, thy sins be forgiven thee.”
Cited for
Jesus could forgive sins because he was God, and his coming death was what paid for them.
The response
Matthew ends the same story with the crowds glorifying God (﷾), "which had given such power unto men". Jesus taught his disciples to ask God (﷾) for the same forgiveness, and said God (﷾) gives it to those who forgive others. When he forgave the paralysed man, and when he taught this prayer, he said nothing of a coming payment. The tax collector in his parable went home justified that same day.
Where it lands — Jesus forgave sins, and taught forgiveness through repentance, long before Christians say his blood was shed.
Romans 3:25 — Epistle to the Romans, chapter 3, verse 25
“Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;”
Cited for
Those forgiven before the cross were forgiven in advance, because of the payment Christ would make. Paul says Christ's blood was for the remission of sins that are past, through the forbearance of God.
The response
Neither Jonah nor Jesus says so. Each gives his own reason. Nineveh was spared because its people turned from their evil way. The tax collector went home justified because he humbled himself.
Where it lands — The Bible gives repentance as the reason for these forgivenesses, not a payment still to come.
Qur'an 22:37 — Surat al-Hajj (The Pilgrimage), verse 37
“It is not their meat nor their blood, that reaches Allah: it is your piety that reaches Him: He has thus made them subject to you, that ye may glorify Allah for His Guidance to you and proclaim the good news to all who do right.”
Cited for
Islam has a blood sacrifice too, at Eid al-Adha and in the Hajj, so blood must matter to God.
The response
Islam does sacrifice an animal at Eid al-Adha and in the Hajj. But the Qur'an says what reaches Allah (﷾): "It is not their meat nor their blood, that reaches Allah: it is your piety that reaches Him." It gives the purpose too: "that ye may glorify Allah for His Guidance to you".
Where it lands — In Islam it is piety, not blood, that reaches God (﷾).
Perspectives
Mainstream Christian
God is holy and just, and sin must be paid for. In love He provided the payment Himself, in the blood of Jesus, so that He can forgive and remain just.
Hebrews 9:22 says that without shedding of blood there is no remission.
Rebuttal
Grant that Hebrews says this. The Law of Moses itself let a man too poor for a lamb or birds bring flour. Of that sin offering it says, "and it shall be forgiven him". Jesus forgave the paralysed man, and God (﷾) forgave Nineveh, with no blood at all.
Paul teaches that believers are justified by the blood of Jesus.
Rebuttal
Grant that Paul, a Christian missionary of the first century, wrote this. But Jesus himself told a paralysed man, "Son, thy sins be forgiven thee". The tax collector who only asked God (﷾) for mercy went home "justified", and neither story mentions blood.
The Islamic perspective
God (﷾) forgives whoever turns to Him in repentance, by His own will and mercy. He needs no blood and no substitute. Jesus taught this, the prophets taught it, and the Qur'an and the Prophet Muhammad (ﷺ) confirm it.
Early in his ministry, Jesus declared a paralysed man's sins forgiven.
Jesus taught his followers to ask God (﷾) directly for forgiveness. He tied it to forgiving others, not to any sacrifice.
In Jesus' parables the lost son and the tax collector are forgiven when they turn back. No one is punished in their place.
Ezekiel promises that the sins of the wicked man who turns back will not be remembered.
God (﷾) spared Nineveh when its people turned from evil, with no sacrifice recorded.
The Law of Moses itself let a poor man's sin be forgiven with an offering of flour, with no blood.
Moses offered himself in Israel's place, but God (﷾) did not accept it. He said the one who sinned would answer for it.
Punishing an innocent in place of the guilty, even a willing one, is not justice. God's (﷾) own rule is that the one who sinned answers for his sin.
God (﷾) can forgive by His will alone, and His own word in Ezekiel says He forgives the one who turns back. That is His justice, not a breach of it.
The Qur'an teaches that Allah (﷾) forgives all sins to those who turn to Him.
The Qur'an teaches that neither the meat nor the blood of the sacrificed animals reaches Allah (﷾). What reaches Him is piety.
The Prophet (ﷺ) reported what Allah (﷾) says of the one who comes to Him with sins that fill the earth. If he did not associate anything with Allah (﷾), Allah (﷾) meets him with as much forgiveness.