Heat olive oil in a large skillet over medium-high heat. Add the onion and cook it until it begins to soften. Then, add the mushroom slices, garlic, and salt, continue cooking until the mushrooms turn lightly brown and the liquid has evaporated. Once done, remove the mixture from the pan and set it aside. Lastly, toss the Mozzarella and Parmesan cheeses together in a small bowl.
Place the first tortilla on the skillet. Sprinkle it with cheese and 1/4 of the mushroom mixture. Add another layer of cheese and place a second tortilla on top. Press down with a spatula to ensure there are no air pockets.
Cook for 1 to 2 minutes, until the bottom tortilla is crisp and brown. Flip the quesadilla over and cook for an additional 1 to 2 minutes until the second tortilla is crisp and brown.
Remove from heat, let it sit for 2 minutes before slicing and serving.