Preheat the oven to 350°F (175°C) and lightly spray an 8-inch square baking dish with cooking spray.
Heat a medium skillet over medium-high heat, add the olive oil, then sauté the finely chopped onion for about 5 minutes until softened.
Reduce the heat to low, then add the cooked quinoa, shredded chicken, black beans, drained green chiles, and enchilada sauce to the skillet; stir until everything is evenly combined and warmed through. Remove from heat.
Stir the shredded cheddar cheese into the quinoa mixture, then transfer the mixture to the prepared baking dish and spread it in an even layer.
Sprinkle the shredded mozzarella evenly over the top.
Bake uncovered for 10 to 15 minutes until the casserole is heated through and the cheese is melted.
Switch the oven to broil and broil 1 to 2 minutes, watching carefully, until the cheese is golden and bubbly; remove from the oven.
Top with dollops of sour cream and sprinkle with chopped cilantro; serve with lime wedges immediately.