Sterilize jars and lids. Put a plate in the freezer. Quarter and core the apples, reserving the cores and seeds. Tie apple trimmings in cheesecloth. Put blackberries and sugar in a wide, 6- to 8-quart preserving pan or other wide, shallow pan. Bring to a simmer over medium-high heat, stirring frequently, and cook until the juices just cover the blackberries, about 5 minutes. Pour into a colander set over a large bowl. Stir berries gently to drain off juice. Return juice to pan, along with the apples and the trimmings. Bring to a boil over high heat and cook, stirring occasionally, until the thickened mixture measures 220 degrees on a candy thermometer, 15 to 20 minutes. Return berries and any juice to pan. Add lemon juice and zest, and bring to simmer. Cook, stirring frequently, about 15 minutes, until a small dab of jam spooned onto the chilled plate and returned to the freezer for a minute wrinkles when you nudge it. Off the heat, discard apples and trimmings. Ladle hot jam into jars, leaving 1/4-inch space at the top. Wipe rims, cover with lids and rings, and tighten until snug. Return jars to canning pot, making sure water covers jars by at least 1 inch. Bring to a boil and boil for 5 minutes. Remove from pot and let jars sit untouched for 12 hours. (After 1 hour, check to see if jars have sealed. Store any unsealed jars in the refrigerator.) Label and store in a cool, dark place.