Neither of those terms is common - having seemingly never come across them in my life until now. Google give about 227k results for "fight academy" as in "Deadeye Dick's Fight Academy" and 102k for the other.
That said, I can't discern any particular rule that would seem to govern except that the examples above all point to the use of a noun, rather than a verb -- music academy, science academy -- so that boxing (and thus fighting) work because they are gerunds, not adjectives in this context.
So: "fighting academy."
Aside: are you teaching "fighting" or something else? Perhaps that other thing should be in the name instead, as both more descriptive and less confusing!