Answer: There are two ways of dealing with the unwanted growth. Cutting it down to ground level and then continually cutting off any sprouts will eventually kill the roots of the plant. There is just so much energy the roots have in store and it takes energy to produce new sprouts. As long as you keep cutting the sprouts down before they produce leaves (which will provide the roots with additional energy), you will eventaully starve out the root system.
The other option is to cut the tree down to ground level and paint the cut stump with full strength Round-Up. Round-Up is a vegetation killer and will be transported to the roots of the tree without harming the flowering quince. Just be careful to keep the Round-Up off your quince.
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