B.B. Warfield, while recognising the obvious fact that the term ‘Trinity’ is not a biblical term, nevertheless maintains that the doctrine of the Holy Trinity is a biblical doctrine.

The doctrine of the Trinity lies in Scripture in solution; when it is crystallized from its solvent it does not cease to be Scriptural, but only comes into clearer view.

The doctrine of the Trinity, insists Warfield, not only is a biblical doctrine; it must be a biblical doctrine, because it is not discoverable by natural reason.  And not only is it not discoverable by human reason, it is not provable from reason.  The presence of various triads in nature and in religion; the various analogies that might be proposed; – none is capable of revealing or proving the Triune nature of God.  To be sure, arguments such as the one that says that because God is love, there must have been an eternal object of his love before anything had yet been created that might have formed that object, are not without value.  But these can at best be regarded as not much more than sign-posts pointing in the direction of the Trinity, rather than proving it.

Is the Trinity clearly revealed in the Old Testament?  Old writers

discovered intimations of the Trinity in such phenomena as the plural form of the Divine name Elohim, the occasional employment with reference to God of plural pronouns (“Let us make man in our image,” Genesis 1:26; 3:22; 11:7; Isaiah 6:8), or of plural verbs (Genesis 20:13; 35:7), certain repetitions of the name of God which seem to distinguish between God and God (Psalm 45:6, 7; 110:1; Hosea 1:7), threefold liturgical formulas Numbers 6:24, 26; Isaiah 6:3), a certain tendency to hypostatize the conception of Wisdom (Proverbs 8), and especially the remarkable phenomena connected with the appearances of the Angel of Jehovah (Genesis 16:2-13, 22:11. 16; 31:11, 13; 48:15, 16; Exodus 3:2, 4, 5; Judges 13:20-22).

And more recent writers appeal to

passages like Psalm 33:6; Isaiah 61:1; 63:9-12; Haggai 2:5, 6, in which God and His Word and His Spirit are brought together, co-causes of effects, are adduced. A tendency is pointed out to hypostatize the Word of God on the one hand (e.g., Genesis 1:3; Psalm 33:6; 107:20; 147:15-18; Isaiah 4:11); and, especially in Ezekiel and the later Prophets, the Spirit of God, on the other (e.g., Genesis 1:2; Isaiah 48:16; 63:10; Ezekiel 2:2; 8:3; Zechariah 7:12). Suggestions ¾ in Isaiah for instance (Isaiah 7:14; 9:6) — of the Deity of the Messiah are appealed to.

Warfield concludes this part of his discussion:-

The Old Testament may be likened to a chamber richly furnished but dimly lighted; the introduction of light brings into it nothing which was not in it before; but it brings out into clearer view much of what is in it but was only dimly or even not at all perceived before. The mystery of the Trinity is not revealed in the Old Testament; but the mystery of the Trinity underlies the Old Testament revelation, and here and there almost comes into view. Thus the Old Testament revelation of God is not corrected by the fuller revelation which follows it, but only perfected, extended and enlarged.

As for the New Testament,

the whole book is Trinitarian to the core; all its teaching is built on the assumption of the Trinity; and its allusions to the Trinity are frequent, cursory, easy and confident…it has been remarked that “the doctrine of the Trinity is not so much heard as overheard in the statements of Scripture.” It would be more exact to say that it is not so much inculcated as presupposed.

So, the Trinity is explicitly revealed neither in the Old Testament nor in the New:-

The Old Testament was written before its revelation; the New Testament after it. The revelation itself was made not in word but in deed. It was made in the incarnation of God the Son, and the outpouring of God the Holy Spirit. The relation of the two Testaments to this revelation is in the one case that of preparation for it, and in the other that of product of it. The revelation itself is embodied just in Christ and the Holy Spirit. This is as much as to say that the revelation of the Trinity was incidental to, and the inevitable effect of, the accomplishment of redemption. It was in the coming of the Son of God in the likeness of sinful flesh to offer Himself a sacrifice for sin; and in the coming of the Holy Spirit to convict the world of sin, of righteousness and of judgment, that the Trinity of Persons in the Unity of the Godhead was once for all revealed to men. Those who knew God the Father, who loved them and gave His own Son to die for them; and the Lord Jesus Christ, who loved them and delivered Himself up an offering and sacrifice for them; and the Spirit of Grace, who loved them and dwelt within them a power not themselves, making for righteousness, knew the Triune God and could not think or speak of God otherwise than as triune.

Biblical and Theological Studies, 22-33