In a small saucepan, bring 3/4 cup water to a boil; add quiona, 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, half the scallions, beans, breadcrumbs, egg, cumin, 1 teaspoon salt, and 1/4 teaspoon pepper; pulse until combined but still slightly chunky.
Form mixture into four 3/4-inch-thick paties (dip hands in water to prevent sticking). If too soft, refrigerate 10 minutes to firm. In a large nonstick skillet, heat oil over medium; cook burgers until browned and cooked through, 8 to 10 minutes per side.
Meanwhile, in a small bowl, combine yogurt, lemon juice, and the remaining scallions; season with salt and pepper. Serve burgers in pita topped with cucumber and yogurt sauce.