The question appears to employ the word 'God' as a name, and not as a definition of a nature, and thus our issue arises. That is to say, the way the word 'God' is here being used would mean to apply it to anyone but the Father—since the Father is God—is to equate and identify that person as the Father, which is by its very nature faulty and wrong. However, this is to import Unitarian theology and understanding of Scripture into a discussion about Trinitarian theology and understanding of Scripture, which will always end in a misunderstanding of the Trinity. When Trinitarians say 'The Son, and the Holy Ghost, these two, not only the Father, are God,' for instance, they aren't using the word 'God' as a name, a single person, but a nature: the Son is truly eternal, omnipotent, etc; likewise the Holy Ghost. This is concordant with the Prologue to St. John's Gospel: "In the beginning was the Word, and the Word was with God, and the Word was God" (John 1:1). It is unadulterated lunacy to say God was with God and was also Himself God (since contextually you have said, "and God was .. θεος." It's not lunacy, but rather the only tenable understanding, to recognize that the Word here said to be with God and also Himself be God, who is said to have become incarnate, referring to Jesus, that two persons are in view, both of whom are God in the aforementioned sense.
The reason the Trinity, like all the mysteries of Christ, were only revealed in the New Testament was on account of the inherent dignity and preciousness of the truths, such that it was fitting that only the Son Himself, the very embodiment of the revelation of the Father, being His very Word, should reveal them: "God, who, at sundry times and in diverse ways, spoke in times past to the fathers by the prophets, last of all, in these days hath spoken to us by his Son, whom he hath appointed heir of all things, by whom also he made the world. " (Heb 1:1-2).
The Athanasian is perhaps the most authoritative and best way to accurately define the Trinity:
But the Catholic faith is this, that we venerate one God in the Trinity, and the Trinity in oneness; neither confounding the persons, nor dividing the substance; for there is one person of the Father, another of the Son, and another of the Holy Spirit; but the divine nature of the Father and of the Son and of the Holy Spirit is one, their glory is equal, their majesty is coeternal.
Of such a nature as the Father is, so is the Son, so also is the Holy Spirit; the Father is uncreated, the Son is uncreated, and the Holy Spirit is uncreated; the Father is infinite, the Son is infinite, and the Holy Spirit is infinite; the Father is eternal, the Son is eternal, and the Holy Spirit is eternal; and nevertheless there are not three eternals but one eternal; just as there are not three uncreated beings, nor three infinite beings, but one uncreated, and one infinite; similarly the Father is almighty, the Son is almighty, and the Holy Spirit is almighty; and yet there are not three almightys but one almighty; thus the Father is God, the Son is God, and the Holy Spirit is God; and nevertheless there are not three gods, but there is one God; so the Father is Lord, the Son is Lord, and the Holy Spirit is Lord; and yet there are not three lords, but there is one Lord; because just as we are compelled by Christian truth to confess singly each one person as God, and also Lord, so we are forbidden by the Catholic religion to say there are three gods or three Lords.
The Father was not made, nor created, nor begotten by anyone. The Son is from the Father alone, not made nor created, but begotten. The Holy Spirit is from the Father and the Son, not made, nor created, nor begotten, but proceeding.
There is, therefore, one Father, not three Fathers; one Son, not three Sons; one Holy Spirit, not three Holy Spirits; and in this Trinity there is nothing first or later, nothing greater or less, but all three Persons are coeternal and coequal with one another, so that in every respect, as has already been said above, both unity in Trinity, and Trinity in unity must be venerated. Therefore, let him who wishes to be saved, think thus concerning the Trinity.
But it is necessary for eternal salvation that he faithfully believes also the incarnation of our Lord Jesus Christ.
Accordingly, it is the right faith, that we believe and confess, that our Lord Jesus Christ, the Son of God is God and man. He is God begotten of the substance of the Father before time, and He is man born of the substance of His mother in time: perfect God, perfect man, consisting of a rational soul and a human body, equal to the Father according to His Godhead, less than the Father according to humanity.
Although he is God and man, yet He is not two, but He is one Christ; one however, not by the conversion of the Divinity into a human body, but by the assumption of humanity in the Godhead; one absolutely not by confusion of substance, but by unity of person. For just as the rational soul and body are one man, so God and man are one Christ.
He suffered for our salvation, descended into hell, on the third day rose again from the dead, ascended into heaven, sits at the right hand of God the Father Almighty; thence He shall come to judge the living and the dead; at His coming all men have to rise again with their bodies and will render an account of their own deeds: and those who have done good, will go into life everlasting, but those who have done evil, into eternal fire.
This is the Catholic faith; unless every one believes this faithfully and firmly, he cannot be saved. Amen.