1. In a small bowl, combine the tomato, chopped onion, lime juice, cilantro, jalapeno, salt and pepper. Set the pico de gallo aside.
2. In a large skillet, heat 1/2 tablespoon olive oil. Add the chicken breast meat, and saute until cooked through and juices run clear. Remove chicken from skillet, and set aside.
3. Put the remaining 1/2 tablespoon of olive oil in the hot skillet and saute the sliced onion and green pepper until tender. Stir in the minced garlic, and saute until the aroma is strong. Finally, mix in half of the pico de gallo and the previously sauteed chicken breast meat. Set this mixture aside; keep warm.
4. In a heavy skillet, heat one flour tortilla. Spread 1/2 of the shredded cheese on the tortilla, and top with the chicken mixture. Sprinkle remaining cheese over the chicken, and top with the remaining tortilla. Flip, and cook on the opposite side. Remove quesadilla from skillet, and cut into quarters. Serve with sour cream and remaining pico de gallo.