No, he will not be fine with broken quills stuck in his back. Even if by some miracle the didn't get terribly infected, they would hurt like hell! They are like little daggers in there. Take that poor dog to the vet.
If you do try to remove them yourself, cut off the tip of the quills to relieve the air pressure in them. If there are a lot of them a vet would be the best route. Quills have a hook on the sharp tip that is stuck inside the skin/meat, so they do not remove easily. At the very least you should call a vet to get advice.