Inspect two 16-ounce jars for cracks and rings for rust, discarding any defective ones. Immerse in simmering water until applesauce is ready. Wash new, unused lids and rings in warm soapy water.

Combine water, quinces, and lemon juice in a pot over medium heat; bring to a boil. Reduce heat; simmer until softened, 8 to 10 minutes. Add apples and sugar; simmer until quinces and apples are very soft, 10 to 15 minutes.

Strain apple-quince mixture through a fine-mesh sieve. Return apple-quince solids to the same pot; heat until hot, stirring constantly.

Pack quince-apple sauce into hot, sterilized jars, filling to within 1/4 inch of the top. Run a clean knife or thin spatula around the insides of the jars to remove any air bubbles. Wipe rims with a moist paper towel to remove any spills. Top with lids and screw rings on tightly.

Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars 2 inches apart into the boiling water using a holder. Pour in more boiling water to cover jars by at least 1 inch. Bring to a rolling boil, cover, and process for 5 minutes.