Preheat the oven to 350 degrees F. Coat a large casserole dish (9x13 inch or similar) with nonstick spray and set aside.
Place 2 cups water and quinoa in a large saucepan, then bring to a low boil. Reduce heat to low, cover, and let cook for 15 minutes. Remove from heat and let stand, covered, for 5 minutes. Fluff with a fork and set aside.
In a Dutch oven or similar large, deep sauté pot, heat the olive oil over medium high. Add the onion, green bell pepper, and red bell pepper. Sauté until slightly softened, about 5 minutes. Add the chicken, chili powder, cumin, salt, black pepper, and cayenne pepper. Continue to sauté until the chicken is cooked through and no longer pink, about 4 to 5 minutes. Add the garlic and drained tomatoes and cook 1 additional minute. Remove from the heat. Stir in the black beans, reserved quinoa, Greek yogurt, 1/4 cup cheddar cheese, and 1/4 cup mozzarella cheese. Taste and adjust seasoning as desired.
Transfer the mixture to the prepared baking dish and spread evenly in the pan. Top with remaining cheddar and mozzarella. Bake, uncovered, until the cheese is melted and the casserole is hot, about 15 minutes. Turn the oven to broil, then continue cooking until the cheese is lightly browned, about 3 to 4 additional minutes, watching carefully so that the cheese does not burn. Let rest 5 minutes, sprinkle with chopped cilantro, and serve.