Set up your Weber Kettle for indirect smoking – Fill your charcoal chimney fire starter with charcoal briquettes, and when ready, empty the chimney in the center of the charcoal grate. With tongs, move the charcoal to both sides of the Weber Kettle, creating two equal piles. Add three more charcoal briquets to each pile.
Bring the chicken leg quarter to room temperature – Pat the chicken wings dry with a paper towel. With your fingers, carefully separate the skin from the chicken without tearing the skin. Place as much pesto under the skin as will fit. With a tablespoon, completely cover the chicken leg quarter with pesto.
Place the chicken leg quarter on the Weber Kettle grill grate – Place the chicken leg quarter in the center of the Weber Kettle grill grate and close the lid. Grill over the indirect heat for 40 minutes depending on the size of your chicken leg quarter, or until a meat thermometer inserted into the thickest part of the thigh reads 165 degrees. Be sure to add more charcoal briquets to both piles if necessary, as the charcoal coals burn down.
Remove the chicken leg quarters – Place the chicken leg quarters on a plate and let rest for 5 minutes. Cut the chicken leg quarter between the drumstick and the thigh and serve.