I can't guarantee this works for quince, but I think it works for most every woody plant. Try painting some Roundup on every stem and sapling that comes up. Eventually, most plants just give up.
Steve's method worked on the flowering quince that was taking over my shrub border. Painting the Roundup on the stump also avoids the damage to other plants when you spray the chemical. It did tak a second application when the plant resprouted.
Is it too big to dig up or pull out completely?
I have had success with the Roundup method as well. I would buy the concentrate and use it full strength, not one of their "ready to use" diluted formulations.
Try drilling hole into the stump and put epsome salt into them. I hear that speeds up the degration process. You can do this in stead of grinding tree stumps.
I agree with Dawn. You can also use road salt. That will kill almost anything.
Wet the stubs and put the cheapest salt you can find on it. My mom had to kill an eighty year old wild rose and it worked. (She had cattle salt, which is not quite as refined as table salt and yet a finer grain than road salt)