The word Trinity comes from the Latin noun Trinitas, meaning the state or condition of being three, or a group of three persons or things. The first recorded application of this Latin word to Father, Son and Holy Spirit was by Tertullian in about 200 where he probably combined Latin words for three and one. Although the word Trinity is not found in Scripture, the principle is:
19 Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit (Matt. 28:19)
The Greek term with the same meaning, Τρίας, has given the English word triad. The Sanskrit word, Trimurti, has a similar meaning.
In view of what is stated about Tertullian, it would be vain to look for the word "Τρίας" (Trinity) in the New Testament, which only speaks of God (often called "the Father"), of Jesus Christ (often called "the Son"), and of the Holy Spirit, and of the relationships between these.
The earliest Christians were noted for their insistence on the existence of one true God, in contrast to the polytheism of the prevailing culture. While maintaining strict monotheism, they developed awareness also that the man Jesus was at the same time something more than a man (an awareness believed to be reflected, for instance, in the opening verses of the Letter to the Hebrews, which describe him as the reflection of God's glory and bearing the impress of God's own being, and, yet more explicitly, in the prologue of the gospel of John) and also with the implications of the presence and power of God that they believed was among them and that they referred to as the Holy Spirit.
In the Beginning Was the Word - John Piper