Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer; cover pan and bring water to a boil. Add beets, cover pan, and steam until just tender, 7 to 10 minutes. Set aside.

Place 2 cups water and quinoa in a saucepan; bring to a boil over high heat. Reduce heat to medium-low, cover, and simmer until quinoa is tender and liquid has been absorbed, about 15 minutes.

Meanwhile, whisk olive oil, red wine vinegar, sugar, garlic, salt, and black pepper together in a large bowl.

Remove quinoa from heat, then immediately add 1/2 of the vinegar dressing while fluffing quinoa with a fork; reserve remaining dressing. Cover and refrigerate quinoa until cool, at least 1 hour.

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