In a medium-size pot, bring 1 1/4 cups water to a boil. Add the quinoa, cover, and reduce the heat to low. Simmer until the quinoa is tender and the liquid has been absorbed, about 12 minutes. Allow to sit, covered, for 5 minutes. Fluff with a fork.
While the quinoa is cooking, whisk olive oil, red wine vinegar, sugar, garlic, salt, and pepper together in a large bowl.
As soon as you have fluffed the hot quinoa, add half of the dressing and toss to evenly distribute; reserve remaining dressing. Cover and refrigerate quinoa until cool, at least 1 hour.
Into the cooled quinoa mixture, stir green onions, arugula, goat cheese (or feta), beets, and additional dressing to moisten. Toss lightly before serving.
Store any leftovers, covered, in the refrigerator and enjoy within the next several days.