To prepare sauce, combine first 3 ingredients, stirring with a whisk; set aside.
To prepare quesadillas, combine peaches, 1 tablespoon chives, and sugar, tossing gently to coat. Heat a large non-stick skillet over medium-high heat. Arrange one-fourth of cheese and one-fourth of peach mixture over half of each tortilla; fold tortillas in half. Coat pan with cooking spray. Place 2 quesadillas in pan; cook 2 minutes on each side or until tortillas are lightly browned and crisp. Remove from pan; keep warm. Repeat procedure with remaining quesadillas. Cut each quesadilla into 3 wedges; serve with sauce. Garnish with chive strips, if desired.