Peel quarter and core quinces, reserving peel and cores, then cut quarters into 1/4 inch pieces. Tie up peel and cores in cheesecloth.
Bring water and sugar to a boil in a 4 quart heavy saucepan, stirring until sugar is dissolved, then add quince and cheesecloth cundle and simmer, partilally covered, until quince is tender, about 1 1/2 hours.
Add cranberries and simmer, uncovered, stirring occasionally, until cranberries burst and soften, 8 to 10 minutes.
Drain mixture in a large medium mesh strainer set over a bowl, discarding cheesecloth bundle and reserving cranberry solids, then return cooking liquid to pan and boil, uncovered until syrupy and reduced to about 1 1/2 cups, 5 to 10 minutes.
Stir together syrup and cranberry mixture and bring to a boil, remove from heat.
Ladle mixture into sterilized jars. Remove air bubbles and fill to 1/4" headspace.
Wipe rims and add hot lid/rings. Process in water bath for 10 minutes once it returns to a rolling boil.