Bring the water to a boil in a large saucepan. Add the quinoa, stir once, and return to a boil. Cook uncovered, over medium heat for 12 minutes. Strain and rinse well with cold water, shaking the sieve well to remove all moisture.
When dry, transfer the quinoa to a large bowl. Add the cucumbers, onion, tomato, parsley, mint, olive oil, vinegar, lemon juice, salt, and pepper and toss well. Spoon onto endive spears, top with avocado, and serve.