Heat oil in large skillet over medium-high heat. Add garlic, onion, bell peppers, mushrooms, jalapeños, salt and pepper. Cook, stirring occasionally, 7-9 minutes or until the vegetables are soft. Transfer to bowl and cool. Stir mayonnaise into cooled mushrooms.
Place tortillas on work surface. Sprinkle with cheese and top with mushroom mixture. Fold each in half to form a semi-circle. Wipe out skillet and return to stove over medium heat. When hot, add quesadillas 2 at a time to skillet. Cook 1-2 minutes per side or until lightly golden and cheese melts. Repeat with remaining quesadillas and serve immediately.
Optional: You can heat in microwave oven if you don`t want your tortillas crispy!