Short answer
No. John 1:1 first says the Word "was with God", so the Word is not the God (﷾) it was with. In the Greek, the second "God" has no article (the word for "the"). The old Greek copies did not mark capital and small letters. So printing "God" or "god" is the translator's choice. The scholars James Moffatt and Edgar Goodspeed wrote "the Logos was divine" and "the Word was divine". The Bible also uses the word god for people. God (﷾) told Moses, "I have made thee a god to Pharaoh". Jesus himself quoted a verse that calls men gods. A title the Bible gives to Moses cannot prove that anyone is the One God. Islam's answer is in the Qur'an, which calls Jesus a word from God (﷾), created by His command "Be".
Full explanation
How Christians read the verse
Christians read the Word as the eternal Son of God. They say he was with the Father before creation and shared his nature. John 1:14 says the Word "was made flesh". Christians take this to mean that God the Son became the man Jesus.
“In the beginning was the Word, and the Word was with God, and the Word was God.”
“And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.”
What the Greek says
In "the Word was with God", the Greek says ton theon, "the God", with the word for "the". In "the Word was God", it has theos with no article. Greek manuscripts of the time made no difference between capital and small letters. So whether to print "God" or "god" is a decision the translator makes from the meaning. The King James Version makes that decision itself in 2 Corinthians 4:4. There Paul uses ho theos, with the article, of Satan. The translators wrote "the god of this world" with a small g. So the article, or the lack of it, does not settle the meaning by itself. The translator's view of the meaning does.
“In the beginning was the Word, and the Word was with God, and the Word was God.”
“In whom the god of this world hath blinded the minds of them which believe not, lest the light of the glorious gospel of Christ, who is the image of God, should shine unto them.”
Translators who wrote "divine"
Two well-known twentieth-century translations by Christian scholars render the clause "the Word was divine" (Goodspeed) and "the Logos was divine" (Moffatt). Logos is the Greek word for "word". Their rendering treats theos without the article as describing what the Word was like. It is not a name for the God (﷾) he was with. Printing a capital G in both places makes the verse say more than the Greek requires.
Moffatt, a Scottish New Testament scholar, translates the last clause of John 1:1 as "the Logos was divine".
Goodspeed, a Greek scholar at the University of Chicago, translates the last clause of John 1:1 as "the Word was divine".
"God" in the Bible's wider use
The Bible uses the word god for more than the One God (﷾). In Hebrew the word is elohim, and in Greek it is theos. God (﷾) made Moses "a god to Pharaoh". The old Greek translation of Exodus 7:1 uses theos there, the very word of John 1:1, without the article. Psalm 82:6 says "Ye are gods". When Jesus was accused of blasphemy (insulting God (﷾)), he answered by quoting that psalm. He said Scripture called them gods "unto whom the word of God came" (John 10:35). So why, he asked, was it blasphemy for him to say he is the Son of God? Jewish tradition took them to be Israel's judges. Thomas also uses the word of Jesus when he says "My Lord and my God" (John 20:28). Muslims read it in the same broad sense the Bible gives the word elsewhere. It is not a claim that Jesus is the One God. In the same chapter, Jesus speaks of "my God, and your God" (John 20:17). Jewish writings from around Jesus' time used the word the same way. Philo, a Jewish thinker of the first century, calls Moses a god. A Dead Sea Scroll (11Q13) applies the word elohim from Psalm 82:1 to Melchizedek, the priest-king who blessed Abram in Genesis.
“And the Lord said unto Moses, See, I have made thee a god to Pharaoh: and Aaron thy brother shall be thy prophet.”
“I have said, Ye are gods; and all of you are children of the most High.”
“Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken; Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God?”
“And Thomas answered and said unto him, My Lord and my God.”
“Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God.”
Philo, a Jewish writer in Alexandria in the time of Jesus, says that Moses was named god and king of the whole nation. He did not mean that Moses was the One God.
A Dead Sea Scroll from the first century BCE, found in Qumran Cave 11. It quotes Psalm 82:1, where the King James Version has "God standeth in the congregation of the mighty", and applies the Hebrew word elohim in that verse to Melchizedek, a heavenly figure who carries out God's judgement, not to the One God.
God's (﷾) word and creation
Christians add that the Word was "in the beginning" and that all things were made by him. They say this is not true of Moses. The Hebrew Bible speaks of God (﷾) creating by His word: "By the word of the Lord were the heavens made". There the word is God's (﷾) command, not a second person beside Him. The Qur'an says Jesus came into being by such a command: God (﷾) said "Be", and he was. So Jesus is what God's (﷾) word brought into being, not the word by which all things were made. In the same Gospel, Jesus calls the Father "the only true God".
“In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made through him. Without him, nothing was made that has been made.”
“By the word of the Lord were the heavens made; and all the host of them by the breath of his mouth.”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
“The similitude of Jesus before Allah is as that of Adam; He created him from dust, then said to him: "Be". And he was.”
“And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
Jesus as a word from God (﷾)
The Qur'an also calls Jesus God's "Word, which He bestowed on Mary". In the same verse it forbids saying "Trinity". Muslim scholars explain that Jesus is called a word because he was created by God's (﷾) word "Be". He was born without a father, as Adam was created without father or mother. A word sent by God (﷾) is not God, just as a messenger is not the one who sent him.
“O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
“The similitude of Jesus before Allah is as that of Adam; He created him from dust, then said to him: "Be". And he was.”
Verse by verse
John 1:1 — Gospel of John, chapter 1, verse 1
“In the beginning was the Word, and the Word was with God, and the Word was God.”
Cited for
"The Word was God": the Word is God by nature, and the Word became flesh in Jesus, so Jesus is God.
The response
The Word was "with God", so it is not the God (﷾) it was with. In the same Gospel, Jesus calls the Father "the only true God". The Greek also allows "divine", as Moffatt and Goodspeed wrote, and the Bible gives the title god to Moses. Jesus himself pointed to those called gods "unto whom the word of God came".
Where it lands — The Word is with God (﷾), not the God (﷾) it is with, and the verse gives him a title the Bible also gives to men. So it does not show that Jesus is the One God.
Perspectives
Mainstream Christian
The Word is the eternal Son, who is God with the Father. John 1:1 calls the Word God, and John 1:14 says the Word became flesh in Jesus.
John 1:1 ends "the Word was God", and John 1:14 says "the Word was made flesh".
Rebuttal
The Greek allows "divine", and Moffatt and Goodspeed wrote it so. The Bible also gives the title god to Moses. John 1:14 says the Word became flesh. It does not say the man is the One God, and in the same Gospel Jesus calls the Father "the only true God".
Christians hold that the Word is a distinct person who shares the one divine nature with the Father.
Rebuttal
The verse says the Word was with God (﷾), so it is not the God (﷾) it was with. Jesus calls the Father "the only true God", which leaves no second person to share that nature. The Qur'an says Jesus was created by God's (﷾) word "Be", so he is not that word but what it brought into being.
The Islamic perspective
John 1:1 does not make Jesus the One God. The Greek allows "divine". The Bible gives the title god to Moses and to men to whom God's (﷾) word came. The Qur'an calls Jesus a word from God (﷾), not God.
In John 1:1 the first theos has the article and the second does not. The Greek manuscripts did not mark capital and small letters, so the capital G is a translator's choice.
Paul uses ho theos of Satan. There the King James translators themselves wrote "the god of this world", with a small g.
Moffatt and Goodspeed, both Christian scholars, translated the clause as "divine". Moffatt has "the Logos was divine", and Goodspeed "the Word was divine".
The Bible calls Moses "a god to Pharaoh", and Psalm 82:6 says "Ye are gods". Jesus quoted the psalm to answer a charge of blasphemy. He explained that Scripture called them gods "unto whom the word of God came".
Philo, a Jewish writer of Jesus' time, calls Moses a god. The Qumran scroll 11Q13 applies elohim from Psalm 82:1 to Melchizedek.
John 1:3 says all things were made through the Word. The Hebrew Bible says God (﷾) made the heavens by His word, His command. In John 17:3 Jesus calls the Father "the only true God". The Qur'an says Jesus was created by that command, "Be".
The Word who "was with God" is not the God (﷾) it was with.
The Qur'an calls Jesus a messenger of Allah (﷾) and His Word bestowed on Mary, created by the command "Be".