Total cooking time for a chicken leg quarter is 45 minutes to an hour whether baked, barbecued or fried. Thicker hind quarters take longer than less meaty specimens. Cooking until an instant-read thermometer inserted in the thickest part of the thigh registers 170 degrees ensures that they are done.
Chicken leg quarters are baked uncovered at 350 degrees, either by themselves or on a bed of onions, potatoes or other vegetables, for 45 minutes to an hour. On the barbecue, the leg quarters are browned over direct heat for about three minutes per side, then moved to the indirect side. The lid is closed or the chicken is tented with foil, and the chicken cooks for 45 minutes more.
To fry, the chicken is browned well over medium-high heat for 15 to 20 minutes on the first side then turned and browned on the other side for another 15 minutes. The heat is then reduced, and the chicken is cooked for an additional 20 minutes.