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.
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 15 minutes.
On the side: 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 cool, at least 1 hour.
Stir green onions, arugula, feta cheese, beets, and remaining dressing into cooled quinoa mixture. Toss lightly before serving.