If you are afraid of losing on the river with a full house Q's over A's in a $5 free roll, you may as well find another hobby. You can be so afraid of losing that you have virtually no chance of winning. Quads happen, but damn seldom.
Yup. It happens very seldom. Unless you have very good reason to think your villain holds quads, it can't be folded. They could also hold a king or another ace.
Calling the river doesn't matter, you have to see the BIG mistake you did in the turn, you have to attack this Ace. Forgetting your turn mistake, I think you should call that river!