Stir salsa, corn, 1/4 cup red onion, and 1/4 cup cilantro in small bowl to blend. Season salsa to taste with salt and pepper.
Sprinkle chicken with chili powder, cumin, salt, and pepper. Heat 2 tablespoons oil in heavy large skillet over medium-high heat. Add garlic and remaining 3/4 cup onion and sauté 1 minute. Add chicken and sauté until chicken is just cooked through, about 5 minutes. Sprinkle half of 1 tortilla with 1/2 cup cheese. Place 1/3 chicken mixture atop cheese; sprinkle with 2 tablespoons cilantro and another 1/2 cup cheese. Fold tortilla in half. Repeat with remaining tortillas, cheese, chicken mixture, and cilantro. Heat 1 tablespoon oil in each of 2 heavy large skillets over medium-high heat. Place 2 quesadillas in 1 skillet and third in second skillet. Cook quesadillas until brown and cheese melts, about 4 minutes per side. Transfer quesadillas to work surface and cut into wedges. Arrange on platter; serve with salsa.