The three most ancient opinions concerning God are Anarchia, Polyarchia, and Monarchia. The first two are the sport of the children of Hellas, and may they continue to be so. For Anarchy is a thing without order; and the Rule of Many is factious, and thus anarchical, and thus disorderly. For both these tend to the same thing, namely disorder; and this to dissolution, for disorder is the first step to dissolution.
But Monarchy is that which we hold in honour. It is, however, a Monarchy that is not limited to one Person, for it is possible for Unity if at variance with itself to come into a condition of plurality; but one which is made of an equality of Nature and a Union of mind, and an identity of motion, and a convergence of its elements to unity— a thing which is impossible to the created nature— so that though numerically distinct there is no severance of Essence. Therefore Unity having from all eternity arrived by motion at Duality, found its rest in Trinity. This is what we mean by Father and Son and Holy Ghost. The Father is the Begetter and the Emitter; without passion of course, and without reference to time, and not in a corporeal manner. The Son is the Begotten, and the Holy Ghost the Emission;
That passage is taken from this Third Theological Oration, and he doesn’t say what he means by it right there. Basil and Nyssa, who are closest to his thought, said it was the hypostasis of the Father (and Eastern Orthodoxy has generally followed that line). However, I read something else by Nazianzen,
But when I say God, I mean Father, Son, and Holy Spirit. For Godhead is neither diffused beyond these, as as to bring in a mob of gods (JA: What Gregory would call anarchia/polyarchia/democracy), nor yet is it bounded by a smaller compass than these, so as to condemn us for a poverty stricken conception of Deity; either Judaizing to save the Monarchia, or falling into heathenism by the multitude of gods. This then is the Holy of Holies (Trinity–JA), which is hidden from the Seraphim…
“On The Theophany, or Birthday of Christ”
Schaff edition, volume 7, p.347
For the underlined part I have the simple definition of monarchia = Father alone = Deity = Judaism, which Gregory rejects. The easiest reading, then, at least in this passage, is that the Monarchia = the Godhead. A lot of interpreters of Gregory (many Roman Catholics and Anglicans, very few EO) have taken that route. This seems to put him at odds with Basil/Nyssa, who said it was the hypostasis of the father. Of course, by identifying the monarchia = the Hypostasis of the Father, they wouldn’t have denied the deity of the other two.