Bring first 7 ingredients to a boil in a saucepan over medium-high heat, stirring often. Reduce heat, and simmer, stirring occasionally, 5 minutes.
Coat food rack with cooking spray; place on grill over medium-high heat (350° to 400°). Arrange chicken quarters on rack, and grill, covered with grill lid, 10 to 15 minutes on each side or until done, basting often with barbecue sauce. (Discard any leftover barbecue sauce used for basting.)
NOTE: Decrease grilling time to 8 to 10 minutes on each side for skinned chicken quarters.