Combine the garlic, lime juice, oil, salt, pepper and sugar. Divide into two bowls and add the chicken, tossing until the chicken is coated. Marinate for at least 4 hours or for up to three days.
Sauté or grill the chicken until it is thoroughly cooked. Cut into thin strips and use immediately, or refrigerate and reheat when ready to serve.
Place a tortilla in the bottom of a no-stick frying pan over medium heat. Sprinkle with cheese, sliced chicken, onions, peppers, tomatoes and hot peppers, if desired. When the cheese begins to melt, fold over the tortilla gently so that the meat and vegetables stay inside. The quesadilla is done when it is lightly browned on both sides.
Cut into quarters and serve as an appetizer or lunch entree with your favorite salsa and Mexican condiments.