In a heavy medium saucepan, combine the apple juice with the sugar and cinnamon stick and bring to a boil, stirring to dissolve the sugar. Add the quinces and cook over moderate heat, stirring occasionally, until tender, about 30 minutes. Add the cranberries and bring to a simmer. Cook until the compote is thick, about 20 minutes. Discard the cinnamon stick before serving.
Make Ahead: The quince compote can be refrigerated in an airtight container for up to 2 weeks.