If the court gave her exclusive use of the home, that does not mean exclusive ownership and does not mean he has to quit claim it.
The only way that happens is if he agrees that her getting the house means he gets his fair share of the equitable distribution through other things they may have.
If they don't own anything worth his share of the home, she could ask the courts to be able to remain in the home until the child graduates high school but he remains part owner. Even though he remains owner, she is responsible for all payments, taxes, insurance, and upkeep. If I were him, I would never agree to this unless she is court ordered to refinance and get his name off the mortgage.
If the mortgage is only 700, I would imagine he may fight her for it.
He has zero reasons to quit claim the property to her and that only means he would be giving up ownership, she can't get his name off the mortgage without refinancing. Would you give up your ownership but remain financially responsible for a property you no longer own??