NP is the set of all functions that you can verify a solution to in polyomial time, and the solution of the inverse-hash function is just a plaintext that hashes to the right value, and obviously you can check if a plaintext is right in polynomial time by just hashing it and comparing the hashes. Thus reversing a hash function is in NP, so if P=NP it's in P.

There's some subtlety here in that "reversing" a hash function really just means coming up with a plaintext that generates the right hash, not the original one, but you can put any polynomial-time set of constraints on the plaintext and finding a plaintext that satisfies those constraints (and hashes to the right value) is still in NP, so the subtlety really doesn't save you much.

Edit:

Side point, but since we're talking quantum, we should really be saying BQP=NP not P=NP, BQP being the problems solvable in polynomial time on a quantum computer, it's a superset of P and a subset of NP, but we don't know if it's equal to either or both. I.e. P=NP implies BQP=NP, BQP != NP implies P != NP, BQP != P implies P != NP, but the reverse of all of those statements isn't known to be true.