A vet told me that if a cat goes outside its litter box, there's a reason.
Here are several possibilities:
If it's a male, get him neutered. If already neutered, feed him low ash food.
Put another litter box in the room where the "accidents on purpose" are happening.
Try a new brand of litter. Sometimes manufacturers change the formula and don't say "New! Improved!"
Good luck! And please let us know which cleaning method worked for the quilt.