Bring vegetable broth, water, and quinoa to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until quinoa is tender and broth has been absorbed, 15 to 20 minutes. Spread quinoa onto a large platter to cool.
Heat 1 tablespoon olive oil in a large skillet over medium heat. Cook and stir onion, zucchini, and carrot in hot oil until the onion is translucent, 3 to 5 minutes; transfer to a plate to cool.
Mix cooled vegetables with 3 cups of the cooled quinoa in a large bowl; season with garlic powder, salt, and black pepper. Pour eggs and ground oats into the bowl; stir until the mixture comes together. Divide mixture into 14 portions and shape into patties.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Fry patties in hot oil until browned, about 2 minutes per side. Use more oil between batches as needed.