Combine the 2 ounces Parmesan with the mozzarella and Cheddar with your fingers in a small bowl and set aside.
Wrap the spinach in paper towels to remove excess moisture. Transfer to a large bowl and add the quinoa, corn and beans. Stir to combine and set aside.
Heat a large nonstick skillet over medium heat.
Meanwhile, place the tortillas on a clean work surface. Working on only the lower halves, divide 1/2 of the cheese blend among the 4 tortillas; top with the spinach-quinoa filling, then sprinkle on the remaining cheese blend. The cheeses are usually salty enough, but season with salt and pepper, if desired. Fold each tortilla in half to make a half-moon.
Spray the preheated skillet with nonstick cooking spray for about 3 seconds to coat. Add 2 of the quesadillas to the skillet and cook until the cheese starts to melt and the tortillas are light brown, about 3 minutes. Transfer the quesadillas, cooked side-down, to a plate. Repeat with the remaining 2 quesadillas.
Sprinkle 1/4 cup of the Parmesan into the skillet in a half-moon shape, then place 1 quesadilla, cooked side-up, directly on the Parmesan to create a crust. Continue to cook until the Parmesan crust browns and the cheese filling is fully melted, about 3 minutes. Remove from the heat and repeat with the remaining Parmesan and quesadillas. Serve immediately or make ahead and keep warm (see Cook's Note).
To make ahead, preheat the oven to 350 degrees F (180 degrees C). Place the cooked quesadillas on a rimmed baking sheet and bake until the cheese filling is warm and gooey.