Heat olive oil in a large nonstick skillet over medium-high heat. Combine ground cumin, 1/8 teaspoon salt, and 1/8 teaspoon black pepper; sprinkle over chicken. Add chicken to pan, and sauté 5 minutes or until browned. Remove chicken from pan; set aside. Add onion, mushrooms, garlic, jalapeño, 1/8 teaspoon salt, and 1/8 teaspoon black pepper to pan; sauté 5 minutes. Remove from pan; let stand 5 minutes. Wipe pan with paper towels.
Heat pan over medium heat. Sprinkle each tortilla with about 1/3 cup cheese. Arrange 1/2 cup mushroom mixture over one-half of each tortilla. Arrange chicken evenly over mushroom mixture. Carefully fold each tortilla in half. Add 2 quesadillas to pan; cook 2 minutes on each side or until lightly browned and cheese melts. Repeat procedure with remaining quesadillas. Serve immediately.