Combine water and quinoa in a pot. Bring to a boil; simmer until water is absorbed, 10 to 15 minutes.
Heat 1 tablespoon oil in a saucepan over medium heat. Add onion and garlic; cook and stir until onion is translucent and soft, about 5 minutes. Add corn, chili powder, cumin, cayenne pepper, salt, and black pepper. Cook and stir until corn is defrosted, about 5 minutes.
Mash beans in a bowl using a fork. Add quinoa and corn mixture; mash until well combined. Add eggs; mix well. Stir in bread crumbs. Form mixture into 4 flat, round patties.
Heat the remaining 1 tablespoon oil in a skillet; cook patties until golden brown, about 4 minutes per side.