Short answer
No. Hebrews 1:8 quotes Psalm 45, a wedding song written for a king of Israel. The very next line says to him, "God, thy God, hath anointed thee". So the one addressed has a God above him, and Hebrews quotes that line too. The Hebrew can also be read "God is thy throne". The Jewish translation of 1917 reads "Thy throne given of God". The Bible also uses the word "god" for people God (﷾) has honoured. God (﷾) told Moses, "I have made thee a god to Pharaoh". A few verses earlier, Hebrews calls Jesus God's Son in words first said of Solomon, who might "commit iniquity". These verses speak of royal honour given by God (﷾), not of a Son who is God. Islam teaches that Jesus was a servant and messenger of God (﷾), honoured by Him.
Full explanation
Why Christians cite the verse
Christians count Hebrews 1:8 among the plainest proof texts. They read it as God the Father speaking to the Son and calling him "God". It comes in a chapter that sets the Son above the angels.
“But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom. Thou hast loved righteousness, and hated iniquity; therefore God, even thy God, hath anointed thee with the oil of gladness above thy fellows.”
The verse it quotes: a song for a king
Hebrews, a letter in the New Testament, quotes Psalm 45:6-7 from the old Greek translation of the Jewish scriptures, with only small differences. The psalm calls itself "a song of loves", spoken "touching the king". It praises him as fairer than other men and blessed by God (﷾). Later it speaks of his queen. The next line after "Thy throne, O God" says to this king, "God, thy God, hath anointed thee". To anoint is to choose and set apart as king. So the one addressed has a God who anoints him, and Hebrews keeps that line.
“To the chief Musician upon Shoshannim, for the sons of Korah, Maschil, A Song of loves. My heart is inditing a good matter: I speak of the things which I have made touching the king: my tongue is the pen of a ready writer. Thou art fairer than the children of men: grace is poured into thy lips: therefore God hath blessed thee for ever.”
“Thy throne, O God, is for ever and ever: the sceptre of thy kingdom is a right sceptre. Thou lovest righteousness, and hatest wickedness: therefore God, thy God, hath anointed thee with the oil of gladness above thy fellows.”
“But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom. Thou hast loved righteousness, and hated iniquity; therefore God, even thy God, hath anointed thee with the oil of gladness above thy fellows.”
Another way to read the Hebrew
The Hebrew line can be read in more than one way. The Jewish Publication Society's 1917 translation reads "Thy throne given of God". The Revised Standard Version (RSV), a modern English Bible, gives "God is thy throne" as an alternative in its note to Hebrews 1:8. It renders the psalm "Your divine throne". On these readings the verse does not call the king God (﷾) at all.
“Thy throne given of God is for ever and ever; A sceptre of equity is the sceptre of thy kingdom.”
The RSV keeps "Thy throne, O God" in the text of Hebrews 1:8 but gives "God is thy throne" as an alternative reading in its footnote. In Psalm 45:6 it translates the line as "Your divine throne endures for ever and ever".
The Bible calls men "god"
Even if "O God" is kept, the title proves nothing on its own. God (﷾) made Moses "a god to Pharaoh". A psalm says to some "Ye are gods". Jesus explained that it called them gods "unto whom the word of God came". He was accused of making himself God, and he answered with that same psalm. If Scripture calls men "gods", he asked, why charge him with blasphemy for saying "I am the Son of God"?
“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?”
"My Son" first meant David's line
Hebrews 1:5 quotes two lines. "Thou art my Son" comes from Psalm 2:7. "I will be to him a Father" comes from 2 Samuel 7:14. Both were first spoken of the kings from David's line. The promise in 2 Samuel is about the son who would build God's (﷾) house. It adds that if he sins God (﷾) will punish him. Chronicles names that son: Solomon.
“For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son?”
“I will declare the decree: the Lord hath said unto me, Thou art my Son; this day have I begotten thee.”
“And when thy days be fulfilled, and thou shalt sleep with thy fathers, I will set up thy seed after thee, which shall proceed out of thy bowels, and I will establish his kingdom. He shall build an house for my name, and I will stablish the throne of his kingdom for ever. I will be his father, and he shall be my son. If he commit iniquity, I will chasten him with the rod of men, and with the stripes of the children of men:”
“Behold, a son shall be born to thee, who shall be a man of rest; and I will give him rest from all his enemies round about: for his name shall be Solomon, and I will give peace and quietness unto Israel in his days. He shall build an house for my name; and he shall be my son, and I will be his father; and I will establish the throne of his kingdom over Israel for ever.”
Words of appointment, not of equality
The chapter's own words describe rank that God (﷾) gives. God (﷾) appointed the Son heir of all things. He was "made" better than the angels, and he "obtained" his name "by inheritance". A God does not inherit his name or get appointed. Hebrews says God (﷾) made the worlds "by" the Son, so even there God (﷾) is the one who acts. Paul, a Christian missionary of the first century, draws the same line. He gives the name "one God", and "of whom are all things", to the Father alone. The Qur'an describes Jesus the same way: a servant of Allah (﷾) to whom God (﷾) gave revelation. Islam teaches that he was a servant and messenger of God (﷾), never God.
“Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds; Who being the brightness of his glory, and the express image of his person, and upholding all things by the word of his power, when he had by himself purged our sins, sat down on the right hand of the Majesty on high; Being made so much better than the angels, as he hath by inheritance obtained a more excellent name than they.”
“But to us there is but one God, the Father, of whom are all things, and we in him; and one Lord Jesus Christ, by whom are all things, and we by him.”
“He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet;”
Verse by verse
Hebrews 1:8-9 — Epistle to the Hebrews, chapter 1, verses 8-9
“But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom. Thou hast loved righteousness, and hated iniquity; therefore God, even thy God, hath anointed thee with the oil of gladness above thy fellows.”
Cited for
The Father speaks to the Son and calls him "God", so the Son is God.
The response
The line is a quotation of Psalm 45, a song for a human king. The next line tells him "God, thy God, hath anointed thee". Hebrews quotes that line too. The Hebrew can also be read "Thy throne given of God".
Where it lands — The verse speaks of someone who has a God, so it cannot make him the one God.
Hebrews 1:10 — Epistle to the Hebrews, chapter 1, verse 10
“And, Thou, Lord, in the beginning hast laid the foundation of the earth; and the heavens are the works of thine hands:”
Cited for
Hebrews applies to the Son a psalm about the Lord laying the foundation of the earth, so the Son is the Creator.
The response
The same chapter says God (﷾) made the worlds through the Son and appointed him heir. The verse just before says "God, even thy God, hath anointed thee". The one addressed has a God above him.
Where it lands — The chapter that quotes the psalm puts God (﷾) above the Son. So it cannot make him the God of Isaiah 44:24, who made the earth "by myself".
Hebrews 1:5 — Epistle to the Hebrews, chapter 1, verse 5
“For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son?”
Cited for
No angel was ever called God's Son like this, so the Son is uniquely divine.
The response
Both quotations in Hebrews 1:5 were first spoken of kings. 2 Samuel 7:14 is about David's son, Solomon, who would build God's (﷾) house. It warns that he may sin. Sonship here is royal honour from God (﷾).
Where it lands — The same words were said of Solomon, and no one takes them to make Solomon God.
Hebrews 1:6 — Epistle to the Hebrews, chapter 1, verse 6
“And again, when he bringeth in the firstbegotten into the world, he saith, And let all the angels of God worship him.”
Cited for
The angels are told to worship the Son, and only God may be worshipped, so the Son is God.
The response
The word "worship" in the Bible also covers the honour paid to a king. In 1 Chronicles, a book of the Jewish scriptures, all the congregation "worshipped the Lord, and the king" in one breath. In the Qur'an, God (﷾) told the angels to bow down to Adam. They obeyed God (﷾), and no one takes it to make Adam God.
Where it lands — The same word is used of honour paid to King David, so it cannot by itself make the Son God.
Perspectives
Mainstream Christian
In Hebrews 1:8 the Father addresses the Son as God. The chapter sets the Son above the angels as the one through whom God made the worlds.
Hebrews 1:8 has God say to the Son, "Thy throne, O God, is for ever and ever".
Rebuttal
Hebrews does quote these words, but from Psalm 45, a song to a human king. The next line says "God, thy God, hath anointed thee", and Hebrews quotes that line too. So the one addressed has a God over him.
Because the Father calls the Son "God", the Son shares the divine nature.
Rebuttal
Grant the title: the Bible calls Moses "a god", and Jesus answered the charge of blasphemy from Psalm 82. A throne "for ever" was promised to Solomon's line too. The next line gives this king a God who anoints him, so the title alone does not show he shares God's (﷾) nature.
The Islamic perspective
Jesus was a servant and messenger of God (﷾). Hebrews 1 quotes royal psalms that honoured Israel's kings, and the very verse it quotes says the one addressed has a God who anointed him.
Hebrews 1:8 quotes Psalm 45, a song addressed to a human king.
The next verse, in both Psalm 45 and Hebrews 1, says that God (﷾), the king's God (﷾), has anointed him. So the one addressed has a God.
The Hebrew can be read "Thy throne given of God" or "God is thy throne". Neither reading says the king is God.
The Bible uses "god" for Moses and for those "unto whom the word of God came". Jesus appealed to this himself.
The words of Hebrews 1:5 were first spoken of David's royal line, including Solomon, who could sin.
The Qur'an teaches that Jesus was a servant of Allah (﷾) and a prophet, not God.