In a small pot, bring 3/4 cup water to a boil. Add quinoa, cover, and reduce heat to low. Cook until liquid is absorbed, 12 to 14 minutes; set aside.
In a food processor, pulse carrot until finely chopped. Add cooked quinoa, scallions, garlic, beans, breadcrumbs, egg, cumin, salt, and pepper; pulse until combined but still slightly chunky. Form the mixture into four even patties.
In a large nonstick skillet, heat the oil over medium; cook burgers until browned, about 8 minutes per side.