The bank doesn't care who owns the property. The bank cares that the mortgage is paid in full. You cannot pay 1/2 of the mortgage -- if you do, the bank will foreclose. You are obligated to pay the full mortgage, because co-borrowers are jointly liable for the entire mortgage.
Your ex was relieved of the obligation to pay on the mortgage via bankruptcy. But, the debt remains, and as the only remaining borrower, you're now stuck with the entire debt. You can either accept the quitclaim, and remain responsible for all of the bills, or you can reject the quitclaim -- but you will still be responsible for all of the bills, because if you don't pay them, the bank, or the county (with regards XXXXX XXXXX taxes), will foreclose.
Note: There is one good reason to reject the quitclaim: as long as your ex remains on title, she is also liable for the property taxes, and you could conceivably sue her for one half of any new property tax bills that accumulate.
Frankly, your best course of action may be to accept the QCD, sell the property, and move on with your life. But, if you're underwater and your loans and back taxes exceed the value of the property, then it really doesn't matter what you do with the QCD. You may as well start considering bankruptcy yourself, because that may be the only way to get free of your obligations on the home.
I realize that this may seem incredibly unfair -- but, FYI, on Jan 1, 2014, the IRS
can start charging taxes on the unpaid balance of any foreclosed property. So, if you really are underwater in this home, then you need to decide if you can sell it before year end, or file bankruptcy. Because if you don't, you may get stuck with a new bill, that you never expected: from the IRS. And, as I'm sure you're aware, Uncle Sam is the one creditor that you do not want to be liable to for anything.
Please let me know if my answer is helpful and if I can be of further assistance.