He may not mention the Holy Spirit in his opening salutations, but he does talk about the Holy Spirit a lot. There are at least 24 mentions of the Holy Spirit in his letters, but he also calls it the Spirit of God, so there are many more. Here are some passages, all the words of Paul.

Acts 20     ²² “And see, now I go bound in the spirit to Jerusalem, not knowing the things that will happen to me there, ²³ except that the Holy Spirit testifies in every city, saying that chains and tribulations await me.

²⁸ Therefore take heed to yourselves and to all the flock, among which the Holy Spirit has made you overseers, to shepherd the church of God which He purchased with His own blood.”

Romans 15     ¹³ Now may the God of hope fill you with all joy and peace in believing, that you may abound in hope by the power of the Holy Spirit.

¹⁵ Nevertheless, brethren, I have written more boldly to you on some points, as reminding you, because of the grace given to me by God, ¹⁶ that I might be a minister of Jesus Christ to the Gentiles, ministering the gospel of God, that the offering of the Gentiles might be acceptable, sanctified by the Holy Spirit.

1 Corinthians 6:19    Or do you not know that your body is the temple of the Holy Spirit who is in you, whom you have from God, and you are not your own?

2 Corinthians 13:14   The grace of the Lord Jesus Christ, and the love of God, and the communion of the Holy Spirit be with you all. Amen.

Ephesians 1:13   In Him you also trusted, after you heard the word of truth, the gospel of your salvation; in whom also, having believed, you were sealed with the Holy Spirit of promise,

1 Thessalonians 1     ⁵ For our gospel did not come to you in word only, but also in power, and in the Holy Spirit and in much assurance, as you know what kind of men we were among you for your sake.

⁶ And you became followers of us and of the Lord, having received the word in much affliction, with joy of the Holy Spirit,

2 Timothy 1:14   That good thing which was committed to you, keep by the Holy Spirit who dwells in us.

Titus 3:5   not by works of righteousness which we have done, but according to His mercy He saved us, through the washing of regeneration and renewing of the Holy Spirit,

Have no fear, Paul recognized the Holy Spirit as much as he did God the Father and Jesus, and he used them all together in some passages. Paul was the theologian of the New Testament, and had no problem with the Holy Spirit being one of the Godhead.

God bless!