In a saucepan, bring the chicken, water and taco seasoning to a boil. Reduce heat; simmer, uncovered, for 10 minutes. Stir in the beans.
Spoon about 1/2 cup chicken mixture over half of each tortilla; top with pimientos, onion, cilantro and cheese. Fold over. In a large skillet over medium heat, cook quesadilla in 1 tablespoon oil for 1-2 minutes on each side or until cheese is melted, using additional oil as needed.