Preheat oven to 350°F.
In a blender or food processor, pulse the chopped broccoli and grated carrot until they are finely chopped but not pureed.
In a large bowl, whisk the eggs lightly.
Add the cooked quinoa, shredded cheddar cheese, the processed broccoli and carrot, and 1/2 teaspoon salt (or omit the salt if serving young children) to the eggs.
Stir the mixture until all ingredients are evenly combined.
Lightly grease two mini muffin tins or spray them with cooking spray. Using a small cookie scoop or spoon, divide the mixture evenly among the mini muffin cups. Press each portion down firmly so the bites hold together.
Bake for 15 to 20 minutes, or until the tops and edges are lightly browned.
Let the quinoa bites cool in the tin for 5 minutes, then remove and serve.