Grill chicken 6 minutes; turn. Add pineapple to barbecue; grill 6 to 7 minutes or until chicken is done (170 degrees F) and pineapple is heated through, turning after 3 minutes. Transfer chicken and pineapple to cutting board.
Reduce barbecue to low heat. Cut chicken into thin strips, then cut pineapple into thin slices.
Sprinkle 2 tablespoons cheese onto half of each tortilla. Top evenly with chicken, pineapple, onions, peppers, barbecue sauce and remaining cheese. Fold tortillas in half; press tops to flatten slightly.
Grill 2 to 3 minutes on each side or until cheese is melted and quesadillas are lightly browned on both sides.