Probably someone has already thought of making modal-ontological arguments for the Trinity, but it is perhaps nevertheless fun to consider the argument(s) below. I was partly inspired by a recent argument of dr. Beau Branson, and moved by talk in Is God the Best Explanation of Things? of an ‘a priori argument for Trinitarianism’ (p.89). Anyhow, here is first a simple argument for the Trinity and then a more elaborate one:

A simple argument

1. Possibly, the Trinity exists.
2. If (1), then the Trinity exists.
3. Therefore, the Trinity exists.

A more elaborate argument

1. A necessarily existing, essentially trinitarian God is metaphysically possible, i.e. exists in at least one possible world.
2. If (1), then a necessarily existing, essentially trinitarian God exists. (by S5 modal logic)
3. Therefore, a necessarily existing, essentially trinitarian God exists.

Of course, an obvious concern is that similar arguments can be run for all sorts of Gods.