If you mean "his/her father", you can only say "apja" or "apukája" or "édesapja" but never "apája." "Apája" is only used by little children maybe when they cannot speak properly yet and you can also find it in some nursery rhymes in this meaning, but that's all. There is another meaning of "apája" though, for example when we don't mean someone's father in particular but we say "az év apája" vagy "a világ legjobb apája" stb.

Well, we usually don't use the word "apája". (Except for children, they sometimes say that.) However, it is not a mistake if you say "apája", it just sounds a little strange.