In a small saucepan, bring water to a boil. Add quinoa and give it a quick stir. Cover the saucepan, reduce heat to low and simmer for 15-16 minutes, or until the quinoa is tender and the water has been absorbed. Remove from heat and let stand 5 minutes. Then fluff with a fork. Allow it to cool to room temperature.
In a large bowl, combine the shredded cooked beet, black olives and cucumber pickles. Stir in quinoa. Add the lemon juice, olive oil, salt and pepper to taste and toss. Taste and adjust seasonings. Serve at room temperature.