The game says it's treated as a Flying/Fighting move, so I'm checking if Flying Gem affects it.
This is because in X/Y, Flying Gem is unobtainable, so we can't test. However this is unlikely, as Flying Press is an attack that is Fighting type, but has the added benefit of damaging with a Flying type identity.
Hope I helped!
I Tested this on Showdown, and it doesnt affect it. http://replay.pokemonshowdown.com/pokemondatabase-customgame-5736
As you can see, Flying Press didn't use up the Flying Gem (if it did it would have said "Hawlucha's Flying Gem activated" or something along those lines).