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, 7 to 10 minutes. Set aside.

Prepare quinoa according to package instructions.

While the quinoa is cooking, whisk olive oil, 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 cool, at least 1 hour.

Stir green onions, arugula, goat cheese, beets, and remaining dressing into cooled quinoa mixture. Toss lightly before serving.