Rinse chicken and pat dry. Trim off excess fat. In an 11- by 16-inch roasting pan (about 3 in. deep), stir wine, shallots, tarragon, salt, and sugar until salt and sugar are dissolved. Add chicken quarters and turn to coat. Cover and chill for 1 day, turning once.
Lift chicken from brine; discard brine. Rinse quarters and pat dry.
Lay chicken quarters on a barbecue grill over medium-hot coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); close lid on gas grill. Cook, turning frequently, until browned on both sides and no longer pink at the bone (cut to test), about 25 minutes. Pile quarters on a platter or place on plates.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.