Preheat oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3 minutes until softened.
Add minced garlic and cook for 1 minute until fragrant.
Add ground turkey to the skillet and cook, breaking it apart with a spatula, until no longer pink, about 6-8 minutes.
Stir in diced red bell pepper, zucchini, and sliced mushrooms. Cook for 5 minutes until vegetables are tender.
Add drained diced tomatoes, tomato sauce, dried oregano, dried basil, salt, black pepper, and crushed red pepper flakes. Stir to combine and simmer for 5 minutes.
Remove skillet from heat and stir in cooked quinoa.
Transfer the mixture to a 9×13 inch baking dish and spread evenly.
Sprinkle shredded mozzarella cheese evenly over the top.
Sprinkle grated Parmesan cheese over the mozzarella.
Cover the baking dish with aluminum foil and refrigerate if making ahead, up to 24 hours.
When ready to bake, preheat oven to 375°F (190°C) if not already heated.
Bake covered for 25 minutes, then remove foil and bake for an additional 15 minutes until cheese is melted and bubbly.
Let the casserole rest for 5 minutes before serving.