Bring the vegetable broth to a boil in a small saucepan and then add the quinoa. Reduce the heat, cover the pan, and simmer the quinoa until fully cooked, about 15 minutes. Then, remove the pan from the heat and let the quinoa steam in the covered pan for 3–5 minutes. Finally, fluff the quinoa with a fork and let it cool to room temperature.
Combine the black beans, garlic, cumin, salt, and pepper in a large bowl and then add the quinoa, corn, bell peppers, and onion. Add the egg and combine the mixture with your hands, mashing a few of the beans as you go to help them create a mixture that can be molded into patties.
Gently form 4 patties, being careful not to pack the mixture too tightly.
Heat the canola oil in a large skillet. When the oil is shimmering, add the patties and cook them until heated through, about 5 minutes per side.
Then, serve the veggie burgers on the toasted buns topped with lettuce, tomato, red onion, and ketchup, to taste.