Short answer
No. The Holy Spirit is not God. Muslim scholars explain that the Qur'an uses this name for the angel Gabriel, an honoured creature who brought God's (﷾) revelation to the prophets. The Hebrew Bible does not make God's (﷾) Spirit a separate person either. There it is His breath and power at work. A Jesuit historian of the Trinity admits that the Jews never saw the Spirit as a person. The one verse that calls the Father, the Word and the Spirit "one" was added to the Bible centuries later. Islam's answer is simple: God (﷾) is One, and Gabriel is His faithful messenger.
Full explanation
What Christians believe
Christians teach the Trinity: one God in three persons, the Father, the Son and the Holy Spirit. They say the Holy Spirit, also called the Holy Ghost, is the third person. They say he is fully God and equal to the Father and the Son. They point to Jesus' command to baptise in the name of the Father, the Son and the Holy Ghost. (To baptise is to wash with water as a sign of joining the faith.) They also point to a verse in the King James Version, an English Bible first printed in 1611. It says the Father, the Word and the Holy Ghost are one. (Christians take "the Word" to mean Jesus.)
“Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:”
“For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.”
The Spirit in the Hebrew Bible
The Hebrew Bible is the Jewish scripture, which Christians call the Old Testament. In it, the Spirit of God (﷾) is God's (﷾) breath and power at work. It moves over the waters, makes the heavens and gives life. The Hebrew word ruach means both breath and spirit. Two historians say that no Old Testament writer treated the Spirit as a separate being. One of them is Edmund Fortman, a Jesuit (a Catholic priest of the Society of Jesus). He wrote a history of the Trinity in 1972. He also notes that Matthew, Mark, Luke and Acts usually present the Spirit as a divine force. Even Luke pairs the Holy Ghost with "the power of the Highest".
“And the earth was without form and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.”
“By the word of the Lord were the heavens made; and all the host of them by the breath of his mouth.”
“The Spirit of God hath made me, and the breath of the Almighty hath given me life.”
“Thou sendest forth thy spirit, they are created: and thou renewest the face of the earth.”
“And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
A Jesuit theologian's history of the doctrine of the Trinity. It states that the Jews never regarded the spirit as a person, that there is no solid evidence any Old Testament writer did (p. 6), and that the Synoptic Gospels and Acts usually present the Holy Spirit as a divine force or power (p. 15).
States that no Old Testament author treated the Holy Spirit as a separate being, the Spirit of God being God's breath that carried His power, and that no New Testament author explained how Father, Son and Spirit relate to each other in essence (p. 62).
The one verse that says the three are one
Only one verse in the Bible says the three are one. It is 1 John 5:7 in the King James Version. Scholars who study the old manuscripts (hand-copied texts) agree that it was added centuries later. It is missing from every known Greek manuscript except eight late ones. Most modern translations leave it out, as the World English Bible, a modern English translation, does here.
“For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.”
“For there are three who testify:”
On 1 John 5:7-8, reports that the words about the Father, the Word and the Holy Spirit are absent from every known Greek manuscript except eight late ones, four of which have them written in the margin as a later addition; that no Greek church father quotes them; and that every ancient translation except the Latin lacks them.
Jesus set the Son and the Spirit apart
Now look at what Jesus himself said. He separates the Son of man (his own title for himself) from the Holy Ghost. A word against the Son of man can be forgiven. A word against the Holy Ghost cannot, in this world or the next. Three equal persons of one God (﷾) would not be treated differently like this.
“Wherefore I say unto you, All manner of sin and blasphemy shall be forgiven unto men: but the blasphemy against the Holy Ghost shall not be forgiven unto men. And whosoever speaketh a word against the Son of man, it shall be forgiven him: but whosoever speaketh against the Holy Ghost, it shall not be forgiven him, neither in this world, neither in the world to come.”
The Holy Spirit in the Qur'an is Gabriel
The Qur'an says that Gabriel brought the revelation down to the heart of the Prophet (ﷺ). In the same way, it says the Holy Spirit brought it. So the two names point to one angel. In Arabic the Qur'an also calls him al-ruh al-amin, the Trustworthy Spirit. (Yusuf Ali, a well-known English translator of the Qur'an, writes "the spirit of Faith and Truth".) It names the Spirit along with the angels, singling him out from among them. It calls the angel sent to Mary ruhana, "Our Spirit" in Arabic. (Yusuf Ali translates it "our angel".) Luke likewise says the angel Gabriel was sent to her. The Qur'an says Jesus was strengthened with the holy spirit. That means an angel supported him. It does not mean he was joined to a third person of God.
“Say: Whoever is an enemy to Gabriel-for he brings down the (revelation) to thy heart by Allah's will, a confirmation of what went before, and guidance and glad tidings for those who believe,-”
“Say, the Holy Spirit has brought the revelation from thy Lord in Truth, in order to strengthen those who believe, and as a Guide and Glad Tidings to Muslims.”
“Verily this is a Revelation from the Lord of the Worlds: With it came down the spirit of Faith and Truth- To thy heart and mind, that thou mayest admonish. In the perspicuous Arabic tongue.”
“Therein come down the angels and the Spirit by Allah's permission, on every errand:”
“She placed a screen (to screen herself) from them; then We sent her our angel, and he appeared before her as a man in all respects.”
“And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth,”
“We gave Moses the Book and followed him up with a succession of messengers; We gave Jesus the son of Mary Clear (Signs) and strengthened him with the holy spirit. Is it that whenever there comes to you a messenger with what ye yourselves desire not, ye are puffed up with pride?- Some ye called impostors, and others ye slay!”
Verse by verse
Matthew 28:19 — Gospel of Matthew, chapter 28, verse 19
“Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:”
Cited for
Baptism in the one name of the Father, the Son and the Holy Ghost shows three persons sharing one divine nature.
The response
Naming three together does not make them one being. Paul, a Christian missionary of the first century, charges Timothy "before God, and the Lord Jesus Christ, and the elect angels". No Christian concludes the angels are God. A Christian reference work that defends the Trinity grants something more. It says this verse, taken by itself, would not decisively prove that the three are persons, or equal, or divine.
Where it lands — The verse names three together. It does not say they are one God or equal.
1 John 5:7 — First Epistle of John, chapter 5, verse 7
“For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.”
Cited for
The Father, the Word and the Holy Ghost bear record in heaven, and these three are one.
The response
These words are missing from every known Greek manuscript except eight late ones. Four of those have them only in the margin. No Greek church father quotes them. Most modern Bibles remove them. The World English Bible reads only "For there are three who testify".
Where it lands — The one explicit verse was added later, so it cannot show what the apostles taught.
Acts 5:3-4 — Acts of the Apostles, chapter 5, verses 3-4
“But Peter said, Ananias, why hath Satan filled thine heart to lie to the Holy Ghost, and to keep back part of the price of the land? Whiles it remained, was it not thine own? and after it was sold, was it not in thine own power? why hast thou conceived this thing in thine heart? thou hast not lied unto men, but unto God.”
Cited for
Lying to the Holy Ghost is lying to God, so the Spirit is God.
The response
Scripture speaks the same way of those through whom God (﷾) acts. Moses told the grumbling Israelites, "your murmurings are not against us, but against the Lord". Paul says that whoever despises his teaching "despiseth not man, but God". To wrong the one who acts for God (﷾) is to wrong God (﷾). That does not make Moses or Paul God.
Where it lands — The verse shows that the Spirit acts for God (﷾). It does not say the Spirit is a separate person who is God.
Perspectives
Trinitarian Christian
The Holy Spirit is the third person of the one God. He is equal to the Father and the Son, and without beginning, like them.
Matthew 28:19 names the Father, the Son and the Holy Ghost together in one baptismal formula. Christians read this as one divine name shared by three persons.
Rebuttal
Naming three together does not make them one being. Paul, a Christian missionary of the first century, charges Timothy "before God, and the Lord Jesus Christ, and the elect angels". No one thinks the angels are God. Even a Christian reference work that defends the Trinity grants this verse alone would not decisively prove the three are persons, equal or divine.
The Islamic perspective
Muslim scholars identify the Holy Spirit as the angel Gabriel. He is an honoured creature who brought the revelation to the prophets and to the Prophet Muhammad (ﷺ). God (﷾) is One, with no partner in His being.
The Qur'an calls the bringer of revelation both Gabriel and the Holy Spirit.
In Arabic the Qur'an calls the Spirit who brought the revelation al-ruh al-amin, the Trustworthy Spirit.
In the Hebrew Bible the Spirit of God (﷾) is His breath and power, not a separate person.
The only verse stating that the three are one is a late addition. It is missing from every Greek manuscript except a few late ones.
Jesus said a word against the Son of man can be forgiven, but not one against the Holy Ghost. So he did not treat them as equal.
The Holy Spirit is an angel of God (﷾) and not God.