Heat olive oil in a saucepan over medium heat; cook and stir onion and garlic until softened, about 2 minutes. Add mushrooms; cook and stir until mushrooms begin to soften, about 1 minute. Fold spinach into mixture and cook until wilted, 3 to 5 minutes. Remove saucepan from heat.
Mix Monterey Jack cheese, cottage cheese, sour cream, salt, and black pepper into spinach mixture.
Heat a skillet over medium heat; warm tortillas, 1 at a time, in the hot skillet until flexible, about 30 seconds per side.
Spoon spinach mixture down the center of each warmed tortilla. Roll tortilla around filling; place filled tortillas seam sides down into a 9x13-inch casserole dish. Pour Alfredo sauce over the enchiladas; top with mozzarella cheese.
Bake in the preheated oven until sauce is bubbling and cheese topping is slightly browned, 20 to 25 minutes.
