Based on your comments, I understand that you are interested if any sort of ontological arguments have been made for the existence of God in a Trinity; and not whether "The Ontological Argument" - usually assumed to mean the medieval argument of Anselm of Canterbury for the existence of God - has been used to prove that God exists in a Trinity.
Oxford dictionaries defines ontology as "the branch of metaphysics dealing with the nature of being". An argument comprises a series of premises intended to support one or more stated conclusions.
With these definitions in mind, an argument for the existence of God as Trinity that perhaps comes the closest to being ontological is that made by Metropolitan Kallistos Ware in his book, The Orthodox Way, where he claims that God's existence as Trinity logically follows from God's nature as being personal and being love. "Both these notions," explains Met. Kallistos, "imply sharing and reciprocity":
First, a “person” is not at all the same as an “individual”. Isolated,
self-dependent, none of us is an authentic person but merely an
individual, a bare unit as recorded in the census. Egocentricity is
the death of true personhood. Each becomes a real person only through
entering into relation with other persons, through living for them and
in them. There can be no man, so it has been rightly said, until there
are at least two men in communication.
The same is true,
secondly, of love. Love cannot exist in isolation, but presupposes the
other. Self-love is the negation of love. As Charles Williams shows to
such devastating effect in his novel Descent into Hell, self-love is
hell; for, carried to its ultimate conclusion, self-love signifies the
end of all joy and all meaning. Hell is not other people; hell is
myself, cut off from others in self-centeredness.
(Kindle Locations 408-414)
"God is far better than the best we know in ourselves," he continues:
If the most precious element in our human life is the relationship of
“I and Thou”, then we cannot but ascribe this same relationship, in
some sense, to the eternal being of God himself. And that is precisely
what the doctrine of the Holy Trinity means. At the very heart of the
divine life, from all eternity God knows himself as “I and Thou” in a
threefold way, and he rejoices continually in this knowledge. All,
then, that is implied in our limited understanding of the human person
and of human love, this we affirm also of God the Trinity, while
adding that in him these things mean infinitely more than we can ever
(Kindle Locations 415-419)
I think Met. Kallistos makes a good philosophical argument for why God must exist in more than one person, but I am not sure we can close the argument that He exists in a Trinity from ontology alone. Why, for example, could God not be a Duality or a Quadrinity? He goes on to write that "the Trinity is not a philosophical theory but the living God whom we worship; and so there comes a point in our approach to the Trinity must give place to wordless prayer." Other knowledge we have of the Trinity comes from truth that has been revealed to us, and is probably not something that we could induce or deduce.