Nooooo, the Holy Spirit is not a dove.
However, in Luke 3:22, the Word says, “And the Holy Spirit descended in bodily form like a dove upon Him…” and, Matthew 3:16 says, “When He had been baptized, Jesus came up immediately from the water; and behold, the heavens were opened to Him, and He saw the Spirit of God descending like a dove and alighting upon Him.
This where the representation of the “dove” originates.
We believe in the Holy Spirit who proceeds from the Father and the Son, and is of the same essential nature, majesty, and glory, as the Father and the Son, truly and eternally God. I Cor. 2:10-12