According to MW online, alliteration is consecutive words beginning with the same (usually) consonant sounds.

So by that definition, neither "happy hour", nor OP's very good attempt to fix it, "appy hour", would count as alliteration.

I always thought repeated words starting with the same vowel sounds would also count. It wouldn't help in this case, though. But apparently repeating vowel sounds is called "assonance."

Never go to a place with a fappy hour, too sticky