Place a steamer insert into a saucepan, and fill with water to just below the bottom of the steamer. Cover pan and bring the water to a boil. Add beets, cover pan and steam until just tender, 15 to 20 minutes. Refrigerate beets until chilled.
Bring quinoa and 2 cups water a saucepan over high heat. Reduce the heat to medium-low, cover, and simmer until the quinoa is tender and the liquid has been absorbed, about 10 minutes.
While the quinoa is cooking, whisk olive oil, red wine vinegar, sugar, garlic, salt and black pepper together in a large bowl.
Remove quinoa from heat, then immediately add half of the vinegar dressing while fluffing the quinoa with a fork; reserve remaining dressing. Cover and refrigerate quinoa until chilled.
Stir green onions, arugula, goat cheese, beets and remaining dressing into cooled quinoa mixture. Toss lightly before serving.