Rinse quinoa under cold water, rubbing grains together; drain.
Heat a medium saucepan over medium-high heat. Add oil to pan; swirl to coat. Add carrot, shallots, and garlic; sauté 2 minutes. Add quinoa; cook 1 minute, stirring constantly. Add 1 1/4 cups water and wine; bring to a boil. Cover, reduce heat, and cook 20 minutes or until liquid is absorbed and quinoa is tender. Remove from heat; fluff with a fork. Stir in thyme, salt, and pepper.