Combine the rice and water in a medium saucepan. Bring to a boil over medium-high heat. Reduce the heat to low, cover, and simmer for 15 minutes. Remove from the heat and let stand for 10 minutes. Fluff with a fork.
Heat the oil in a large skillet over medium heat. Add the onion, bell pepper, and carrot; cook until softened, about 5 minutes. Add the cooked rice and stir to combine.
In a small bowl, whisk together the egg, salt, and pepper until well blended. Pour into the skillet with the rice mixture; cook until heated through.
In a shallow dish or pie plate, spread out the flour. Scoop up about 1 tablespoon of the rice mixture at a time; shape into balls using your hands. Roll each ball in the flour to coat lightly.
Heat 1/4 inch of oil in a large skillet over medium-high heat until hot but not smoking. Fry the rice balls in batches until golden brown on all sides; drain on paper towels.