Assuming your name was not on the property deed prior to the quit-claim, your mom gifted the property to you. So your cost basis, is whatever your mom's cost basis was, plus the cost of any property improvements you paid for after you owned it.
So your cost basis is what your mom paid for it, plus the cost of any improvements your mom paid for while she owned it, plus the cost of any improvements you paid for while you owned it. So if you sold it for more than your final cost basis, you pay taxes on the difference.
