Melt butter in a large skillet over medium-low heat; cook and stir onion until softened and lightly browned, about 10 minutes. Add jalapeno peppers to onion; cook and stir until onion is browned, about 10 minutes more. Transfer onion mixture to a plate.

Heat about 1 tablespoon olive oil in the same skillet and add 1 tortilla. Top tortilla with 1/4 cup Mexican cheese blend, 1/2 of the onion mixture, and 1/4 cup Mexican cheese blend, respectively; top with 1 tortilla. Cook quesadilla until browned and cheese is melted, 1 to 2 minutes per side. Repeat with remaining oil, tortillas, cheese, and onion mixture.