Scriptures mention the Father and Son so that people wouldn't deny the Son.
People should not deny the Holy Spirit either.
The Holy Spirit existed from the beginning. He is often called the Spirit of God or the Spirit of Jesus in the Bible.
"In the beginning God created the heavens and the earth. And the earth was a formless and desolate emptiness, and darkness was over the surface of the deep, and the Spirit of God was hovering over the surface of the waters. Then God said, ‘Let there be light’; and there was light…" (Genesis 1:1-3).
This Holy Spirit was actually in the Temple of God in Jerusalem during Old Testament times (Ezekiel 43:5).
The Holy Spirit has been mentioned numerous times throughout the Bible.
Jesus said: "I will ask the Father, and He will give you another Helper, that He may be with you forever; that is the Spirit of truth, whom the world cannot receive, because it does not see Him or know Him, but you know Him because He abides with you and will be in you." (John 14:16-17)
"But know this first of all, that no prophecy of Scripture is a matter of one’s own interpretation, for no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God." (2 Peter 1:20-21)
"And God, who knows the heart, testified to them giving them the Holy Spirit, just as He also did to us; and He made no distinction between us and them, cleansing their hearts by faith." (Acts 15:8-9)
"And Jesus came up and spoke to them, saying, 'All authority has been given to Me in heaven and on earth. Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age.'" (Matthew 28:18-20)
When you believe in Jesus, the Holy Spirit dwells in you.
“And you also were included in Christ when you heard the message of truth, the gospel of your salvation. When you believed, you were marked in Him with a seal, the promised Holy Spirit, who is a deposit guaranteeing our inheritance until the redemption of those who are God’s possession.” (Ephesians 1:13-14)
"However, you are not in the flesh but in the Spirit, if indeed the Spirit of God dwells in you. But if anyone does not have the Spirit of Christ, he does not belong to Him." (Romans 8:9)