The Bible does not specifically talk about the theology of the Holy Trinity, as such. But there is more to say than that. It does form the basis for the developed theology of the Holy Trinity as we know it today. In Judaism there was plenty of background to speak of God the Father as well as to speak of the Spirit of God. With the coming of Jesus, the idea of the Son of God completed the trio. Next would come people wondering just how all of that stuff worked out. If God is One, how can we have a Father, a Son, and a Spirit? Over time, as the books of the New Testament were written those questions were asked and explored.
In the Old Testament, some writers found clues. The first-person plural pronouns in Genesis 1:26–27 and Genesis 3:22 have been used to argue for a Trinitarian understanding of God: “Then God said, ‘Let us make man in our image, after our likeness” and “Then the Lord God said, “Behold, the man has become like one of us in knowing good and evil.” St. Augustine of Hippo, held that the three visitors to Abraham were the three persons of the Trinity. (Gen 18:1-2, And the Lord appeared to him by the oaks of Mamre, as he sat at the door of his tent in the heat of the day. He lifted up his eyes and looked, and behold, three men were standing in front of him.)
Although the doctrine of the Trinity is not explicit in the books of the New Testament, there is an understanding of God that is expressed and contains a number of what are called Trinitarian formulas. If you want to look some of them up try; Matthew 28:19, 2 Corinthians 13:14, Ephesians 4:4–6, 1 Peter 1:2, and Revelation 1:4–6.
These passages include the Great Commission and Paul the Apostle’s blessing, which are quoted below:
Matt 28:18-20: And Jesus came and said to them, “All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.” ESV
2 Cor 13:14: The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all. ESV
There is one case in this discussion that bears special mention. It occurs only in the King James Version because of a difficulty that the translators found themselves in. The King had specified that this translation of the scriptures into English must be from the Hebrew for the Old Testament and the Greek for the New Testament. In 1611, when the translations were done there were very few ancient manuscripts of the scriptures in those original languages and the one Greek New Testament was a hand-copied and bound volume that did not agree with other texts. This was not uncommon since before the invention of the printing press all copies of the bible were hand copied. A part of the book of 1 John that was not in agreement was therefore taken from the Latin Vulgate, translated into Greek and then was translated into English. That portion reads: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. KJV
Verse 7 is known as the Johannine Comma, which most scholars agree to be a later addition by a later copyist or what is termed a textual gloss (or footnote) and was not part of the original text. This verse read in 1611:
For there are three that beare record [in heaven, the Father, the Word, and the Holy Ghost: and these three are one.] [And there are three that beare witnesse in earth], the Spirit, and the Water, and the Blood, and these three agree in one.— 1 John 5:7
This verse is absent from the Ethiopic, Aramaic, Syriac, Slavic, early Armenian, Georgian, and Arabic translations of the Greek New Testament. It is primarily found in Latin manuscripts, although a minority of Greek, Slavonic, and late Armenian manuscripts contain it.
I like to think that God brought a Bedouin shepherd to an ancient cave and caused him to find what we know as The Dead Sea Scrolls in order to provide us with a trove of ancient documents so that mankind can solve the difficulties caused by so many centuries of human error.