No where in the Bible does it say that the Holy Spirit is Gabriel, but it says in a number of places that the Holy Spirit is part of Godhead
1 John 5:7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
In addition, Jesus talked of the Comforter
John 14:26 But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
Acts 1:7 And he said unto them, It is not for you to know the times or the seasons, which the Father hath put in his own power. 8 But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judæa, and in Samaria, and unto the uttermost part of the earth.
Acts 2:1 And when the day of Pentecost was fully come, they were all with one accord in one place. 2 And suddenly there came a sound from heaven as of a rushing mighty wind, and it filled all the house where they were sitting. 3 And there appeared unto them cloven tongues like as of fire, and it sat upon each of them. 4 And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance.
Compare these items to those that mention Gabriel:
Luke 1:19 And the angel answering said unto him, I am Gabriel, that stand in the presence of God; and am sent to speak unto thee, and to shew thee these glad tidings.
Luke 1:26 And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth,
Daniel 8:15 ¶ And it came to pass, when I, even I Daniel, had seen the vision, and sought for the meaning, then, behold, there stood before me as the appearance of a man. 16 And I heard a man’s voice between the banks of Ulai, which called, and said, Gabriel, make this man to understand the vision. 17 So he came near where I stood: and when he came, I was afraid, and fell upon my face: but he said unto me, Understand, O son of man: for at the time of the end shall be the vision.
Bottom line – there are all sorts of references to the Holy Spirit as part of the trinity Godhead and part of God, and there are no references to the Holy Spirit being an angel. On the other hand, Gabriel is no doubt an angel, and appears as an entity that is separate from God and bears messages from God.