Short answer
No. Jesus told a paralysed man that his sins were forgiven. Some scribes (Jewish teachers of the law) said only God (﷾) can forgive sins. But the Gospels answer them. Matthew ends the story with the crowd praising God (﷾), who had given such power to men. In John, Jesus gives his disciples the same power, and no church says that made them God. Jesus also said he could do nothing of his own self. The prophet Nathan told King David that God (﷾) had put away his sin. And Jesus forgave this man long before the cross Christians speak of, so his forgiveness needed no sacrifice. Islam agrees that only God (﷾) forgives. The Qur'an asks: who can forgive sins except Allah (﷾)? Jesus announced God's (﷾) forgiveness as His messenger.
Full explanation
Why Christians see a claim to be God
In Mark 2 Jesus tells a paralysed man that his sins are forgiven. Some scribes, the Jewish teachers of the law, call this blasphemy (speaking against God (﷾)), because only God (﷾) forgives sins. Jesus answers by healing the man, so that they may know the Son of man has power on earth to forgive sins. Son of man is the title he uses for himself. Luke tells of a second time, when the guests ask who this is that forgives sins. Christians read these scenes as Jesus using God's (﷾) own authority.
“When Jesus saw their faith, he said unto the sick of the palsy, Son, thy sins be forgiven thee.”
“Why doth this man thus speak blasphemies? who can forgive sins but God only?”
“But that ye may know that the Son of man hath power on earth to forgive sins, (he saith to the sick of the palsy,)”
“And he said unto her, Thy sins are forgiven. And they that sat at meat with him began to say within themselves, Who is this that forgiveth sins also?”
Matthew: a power God (﷾) gave to men
Matthew tells the same healing, with Jesus' key sentence in almost the same words. But he ends it with the crowd's verdict, not the scribes' charge. They glorified God (﷾), who had given such power to men. Jesus says the same of all his authority: it was given to him.
“But that ye may know that the Son of man hath power on earth to forgive sins, (then saith he to the sick of the palsy,) Arise, take up thy bed, and go unto thine house.”
“But when the multitudes saw it, they marvelled, and glorified God, which had given such power unto men.”
“And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth.”
The disciples were given the same power
In John's Gospel Jesus sends his disciples as the Father sent him, and tells them that the sins they forgive are forgiven. If forgiving sins proved that Jesus was God, it would prove the same of the disciples. No Christian church draws that conclusion. The power is received and passed on, which is the mark of a messenger.
“Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you.”
“Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.”
What Jesus said of his own power
Jesus says plainly that he can do nothing of his own self. He judges as he hears, and seeks the will of the Father who sent him. Whatever he said about forgiveness, he said as one sent.
“I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me.”
A prophet who announced God's (﷾) forgiveness
The Hebrew Bible already shows a prophet telling a sinner that his sin is forgiven. When David confessed, the prophet Nathan told him that the Lord had put away his sin. Nobody reads this as a claim that Nathan was God. He was announcing what God (﷾) had done, and Jesus did the same.
“And David said unto Nathan, I have sinned against the Lord. And Nathan said unto David, The Lord also hath put away thy sin; thou shalt not die.”
Forgiveness before any talk of a cross
Jesus forgave the paralysed man, and the woman in Luke, long before the cross Christians speak of. No death was needed for their forgiveness. He taught his disciples to ask God (﷾) for forgiveness, and said the Father forgives those who forgive others. There is no word of blood or sacrifice in it.
“When Jesus saw their faith, he said unto the sick of the palsy, Son, thy sins be forgiven thee.”
“And he said unto her, Thy sins are forgiven. And they that sat at meat with him began to say within themselves, Who is this that forgiveth sins also?”
“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.”
What the Qur'an says
The Qur'an asks who can forgive sins except Allah (﷾), and promises that Allah (﷾) forgives all sins to those who turn back to Him. It describes the miracles of Jesus as done by Allah's (﷾) leave. So Muslims hold that Jesus healed by Allah's (﷾) leave and announced forgiveness as a messenger. The forgiving was God's (﷾).
“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.”
“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.”
“"And (appoint him) a messenger to the Children of Israel, (with this message): "'I have come to you, with a Sign from your Lord, in that I make for you out of clay, as it were, the figure of a bird, and breathe into it, and it becomes a bird by Allah's leave: And I heal those born blind, and the lepers, and I quicken the dead, by Allah's leave; and I declare to you what ye eat, and what ye store in your houses. Surely therein is a Sign for you if ye did believe;”
Verse by verse
Mark 2:7 — Gospel of Mark, chapter 2, verse 7
“Why doth this man thus speak blasphemies? who can forgive sins but God only?”
Cited for
Only God can forgive sins. Jesus forgave sins, and the scribes rightly saw that he was claiming to be God.
The response
This is the scribes' objection, not Mark's verdict. Matthew tells the same story and gives the verdict of the crowd: they glorified God (﷾), who had given such power to men.
Where it lands — Matthew closes the story with the crowd glorifying God (﷾) for giving such power to men, not with the scribes' charge.
John 20:23 — Gospel of John, chapter 20, verse 23
“Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.”
Cited for
The disciples only declare forgiveness on Christ's behalf. Jesus forgave on his own authority, which is different.
The response
Jesus says his own authority was given to him too: all power is given unto me, and I can of mine own self do nothing. He sends the disciples as the Father sent him. At each step the power is received from God (﷾).
Where it lands — Jesus and the disciples both forgave with a power they were given, so forgiving sins does not prove that either is God.
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 cross to come. Christians point to Romans 3:25.
The response
Neither Mark nor Jesus says that these people were forgiven in advance because of the cross. Jesus gives his own reason for God's (﷾) forgiveness: if ye forgive men their trespasses, your heavenly Father will also forgive you. He also told of a tax collector who prayed, God (﷾) be merciful to me a sinner. That man went home justified because he humbled himself. No cross is mentioned.
Where it lands — In every forgiveness the Gospels actually show, Jesus tied it to faith, humility and forgiving others, and none of them waited for a cross.
Perspectives
Mainstream Christian
Only God can forgive sins committed against God. Jesus forgave sins on his own authority, the scribes understood it as a claim to God's prerogative, and he proved his authority by healing.
In Mark 2 the scribes object that only God can forgive sins. Jesus answers that the Son of man has power on earth to forgive sins.
Rebuttal
Islam agrees that only God (﷾) forgives sins, but the scribes' objection is not Mark's verdict. Matthew ends the same story with the crowd glorifying God (﷾) for giving such power to men. Jesus himself says all power was given to him.
Christians read Jesus' forgiving of sins in his own name as a claim to divine authority.
Rebuttal
In these scenes Jesus never says "I forgive you". He says "thy sins be forgiven thee", as Nathan told David the LORD had put away his sin. And he sends his disciples to forgive in the same way, and he says he can do nothing of his own self.
The Islamic perspective
Jesus announced God's (﷾) forgiveness as a prophet, with a power God (﷾) gave him. He passed the same power to his disciples, so it cannot prove that he is God. Only Allah (﷾) forgives sins.
Matthew ends the healing of the paralysed man with the crowd glorifying God (﷾). He had given such power to men.
In John 20:23 Jesus gives his disciples authority to forgive sins, and no church holds that this made them God.
Jesus said he could do nothing of his own self, and that all power was given to him.
The prophet Nathan announced God's (﷾) forgiveness to David, as Jesus did to the paralysed man.
Jesus forgave sins long before the cross Christians speak of. He taught that the Father forgives those who forgive others, with no sacrifice named.
The Qur'an teaches that none can forgive sins except Allah (﷾). Jesus worked his miracles by Allah's (﷾) leave.