Not only is PA not finitely axiomatizable, but any consistent extension of PA isn't (I think this is true, the same proof that works for PA should go through here, but I haven't checked the details)

Well if we had this, we would know immediately that Q + Con(PA) is not an extension of PA (which is what we originally wanted), because it certainly is finitely axiomatizable. I know there are several proofs that PA is not finitely axiomatizable, but I have not read any of them, so can't comment on the strengthened statement, though it sounds true.