Heat a large skillet over medium-low heat. Cook apple, corn syrup, honey, maple syrup, ground cinnamon, and brown sugar in hot skillet, stirring every few minutes, until apples are slightly softened and caramelized, 8 to 10 minutes. Transfer apple mixture to a bowl.
Place 1 tortilla in the same skillet over medium-low heat; spread apple mixture on top of tortilla. Top with remaining tortilla. Cook until the bottom tortilla is golden brown, 1 to 3 minutes. Carefully flip the quesadilla and cook until the other tortilla is golden brown, 1 to 3 minutes more.