Boil them in a large, tightly-covered pot in ½ cup water and ½ cup of lemon juice, on moderate heat, for about ½ an hour, or until tender, then push them through a coarse sieve, or squish in your hands, about a cup at a time, discarding the hard cores, etc. Puree until very smooth in a food processor or with a wand.
OR Microwave the halved or quartered quinces in a covered microwave bowl in ¼ cup each water and lemon juice, for about 15 minutes or until the flesh (but not the cores) are tender. Discard the cores and puree as above.
Measure the volume of the puree and put it in a pot with an equal volume of sugar. (Do not over-fill the pan, but cook in two batches if necessary.) Wearing a heavy rubber glove to prevent burns from popping puree, stir the mixture regularly as it cooks on moderate heat, thickens and darkens in colour. This should take up to half an hour. Left unstirred, the mixture may darken too much, stick to the bottom, and burn.
The mixture is ready when a little on a cold plate sets hard after standing in a cool place for several minutes. The colour varies. Sometimes it is darker than others - perhaps this has to do with the ripeness of the quinces.