Holy Spirit is God

Muslims believe that the Holy Spirit in the Bible is the angel Gabriel. 

Why Muslims believe the Holy Spirit is the angel Gabriel.

The Holy Spirit described in the Bible is not God but the angel Gabriel. In the Quran, the angel Gabriel is referred to as the Holy Spirit.

When Allah says : O Jesus, son of Mary! Remember My favor to you and to your mother; how I strengthened you with the Holy Spirit..(Surah 5:110)

Why Christians believe the Holy Spirit cannot refer to the angel Gabriel.

The Bible describes the Holy Spirit and the angel Gabriel as two distinct persons

(Luke 1:13-19 NIV)  But the angel said to him: "Do not be afraid, Zechariah; your prayer has been heard. Your wife Elizabeth will bear you a son, and you are to give him the name John. {14} He will be a joy and delight to you, and many will rejoice because of his birth, {15} for he will be great in the sight of the Lord. He is never to take wine or other fermented drink, and he will be filled with the Holy Spirit even from birth…. {18} Zechariah asked the angel, "How can I be sure of this? I am an old man and my wife is well along in years." {19} The angel answered, "I am Gabriel. I stand in the presence of God, and I have been sent to speak to you and to tell you this good news.

It is clear the angel Gabriel was not talking about himself when he announced that John the Baptist will be filled with the Holy Spirit from birth.

(Luke 1:26-35 NIV)  In the sixth month, God sent the angel Gabriel to Nazareth, a town in Galilee, {27} to a virgin pledged to be married to a man named Joseph, a descendant of David. The virgin's name was Mary. {28} The angel went to her and said, "Greetings, you who are highly favored! The Lord is with you." {29} Mary was greatly troubled at his words and wondered what kind of greeting this might be. {30} But the angel said to her, "Do not be afraid, Mary, you have found favor with God. {31} You will be with child and give birth to a son, and you are to give him the name Jesus. {32} He will be great and will be called the Son of the Most High. The Lord God will give him the throne of his father David, {33} and he will reign over the house of Jacob forever; his kingdom will never end." {34} "How will this be," Mary asked the angel, "since I am a virgin?" {35} The angel answered, "The Holy Spirit will come upon you, and the power of the Most High will overshadow you. So the holy one to be born will be called the Son of God.

Likewise in the above verses, the angel Gabriel and the Holy Spirit are two distinct persons.

The Bible teaches that the Holy Spirit is God

(Acts 5:3-4)  Then Peter said, "Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? {4} Didn't it belong to you before it was sold? And after it was sold, wasn't the money at your disposal? What made you think of doing such a thing? You have not lied to men but to God."

The description of the Holy Spirit fit the description of God.

When the Bible talks of the Holy Spirit as God, we know from the context that it is referring to the almighty God.

Omnipresent

(John 14:16-17) And I will ask the Father, and he will give you another Counselor to be with you forever-- {17} the Spirit of truth. The world cannot accept him, because it neither sees him nor knows him. But you know him, for he lives with you and will be in you.

The Holy Spirit is an omnipresent being that will live within each believer.

(Psa 139:7-9) Where can I go from your Spirit? Where can I flee from your presence? {8} If I go up to the heavens, you are there; if I make my bed in the depths, you are there. {9} If I rise on the wings of the dawn, if I settle on the far side of the sea,

{C}2{C}High in rank

The Holy Spirit is so high in rank that lying to him requires the punishment of instant death.

(Acts 5:3-4) Then Peter said, "Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? {4} Didn't it belong to you before it was sold? And after it was sold, wasn't the money at your disposal? What made you think of doing such a thing? You have not lied to men but to God."

Jesus taught that blasphemy against the Holy Spirit is so serious, it will not be forgiven.

(Mat 12:31) And so I tell you, every sin and blasphemy will be forgiven men, but the blasphemy against the Spirit will not be forgiven.

Named alongside the persons of the Godhead.

Jesus also saw it fit to include the name of the Holy Spirit during baptism alongside the name of the Father.

(Mat 28:19) Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,

The Holy Spirit is also involved with the other person's of the Godhead in God's redemptive plan of salvation.

(John 3:5) Jesus answered, "I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit.

The Holy Spirit is only described as “God” in the metaphorical sense.

There has been many uses of the word "god" in the metaphorical sense. Angels were referred to as gods.

(Psa 8:5) For thou hast made him (ie Jesus) a little lower than the angels (same word translated gods in the two verses above), and hast crowned him with glory and honour.

The judges of Israel, who were human, were also called "gods".

(Psa 82:1-6) God presides in the great assembly; he gives judgment among the "gods": {2} "How long will you defend the unjust and show partiality to the wicked? …. {6} "I said, 'You are "gods"; you are all sons of the Most High.' (also refer to John 10:34-35)

Evil spirits were also referred to as gods.

(Jer 10:11) "Tell them this: 'These gods, who did not make the heavens and the earth, will perish from the earth and from under the heavens.'"

Therefore, the fact that the Holy Spirit is referred to as God by Jesus' disciples does not prove that they believe that he is equal to God the Father.

What Christians say

Muslims are right in saying that the term "god" can be metaphorically applied to beings other than God. However when such usage occurs, it is clear from the context that the "god" spoken of is not referring to God. For example, no one who reads Psalms 82 thinks that it is referring to God Almighty. The chapter is actually a rebuke to the judges (gods).

Are the Holy Spirit and God separate persons?
Muslims believe that the Holy Spirit may just be a term to refer to God the Father.

The Holy Spirit may also be a term used to describe God. The Old Testament used the term "Spirit" even though his existence was not revealed until Jesus taught his disciples about the Holy Spirit.

(Gen 1:1-2) In the beginning God created the heavens and the earth. {2} Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters.

(Psa 139:7) Where can I go from your Spirit? Where can I flee from your presence?

What Christians say

It is possible, even likely that the writer of the Old Testament, writing under God's inspiration, included references to the Holy Spirit, even though it was not until Jesus that the Holy Spirit was introduced in greater detail. This kind of revelation is no different from writers who penned prophetic passages under God's inspiration, not knowing exactly what was being written as the prophesy was fulfilled much later.

The Holy Spirit and the Father are separate persons

Jesus made it clear that the Father and the Holy Spirit are two distinct persons. The names of three distinct persons are invoked during baptism.

(Mat 28:19) Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,

The distinctness of the Holy Spirit from the Father and Son can also be seen in the scene right after Jesus' baptism by John the Baptist.

(Luke 3:22) and the Holy Spirit descended on him in bodily form like a dove. And a voice came from heaven (the Father's voice): "You are my Son, whom I love; with you I am well pleased."

Jesus further described the Holy Spirit as a person, distinct from the Father, whom the Father will send. The Holy Spirit is a person and is addressed with the pronoun "he".

(John 14:26) But the Counselor, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.