Preheat the oven to 375°F (190°C).
In a medium saucepan, combine the rinsed quinoa and chicken broth. Bring to a boil over medium-high heat, then reduce heat to low, cover, and simmer for 15 minutes or until the quinoa is cooked and liquid is absorbed. Remove from heat and fluff with a fork.
While the quinoa cooks, heat olive oil in a large skillet over medium heat. Add the diced onion and sauté for 3-4 minutes until softened.
Add the minced garlic to the skillet and cook for an additional 1 minute until fragrant.
Add the chopped broccoli florets and shredded carrots to the skillet. Cook for 5 minutes until vegetables are tender-crisp. Remove from heat.
In a large mixing bowl, combine the cooked quinoa, sautéed vegetables, shredded chicken, shredded cheddar cheese, shredded mozzarella cheese, Greek yogurt, milk, dried Italian seasoning, salt, black pepper, and smoked paprika. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13-inch casserole dish and spread evenly.
Bake uncovered in the preheated oven for 25 minutes, or until the casserole is bubbly and the cheese is melted and slightly golden on top.
Remove from oven and let the casserole rest for 5 minutes before serving.