Bring 1 3/4 cups water, the quinoa and a pinch of salt to a boil in a medium saucepan. Reduce the heat to medium low; cover and cook until the quinoa is tender and the liquid is absorbed, 10 to 15 minutes. Remove from the heat and fluff with a fork; let cool slightly.
Meanwhile, toss the chickpeas, cucumber, tomatoes, red onion, vinegar, 2 tablespoons olive oil and the oregano in a large bowl; season with salt and pepper and let sit 10 minutes.
Pulse the feta, 2 tablespoons water and the remaining 1 tablespoon olive oil in a mini food processor until smooth, adding up to 1 tablespoon water, if needed; season with salt and pepper.
Divide the lettuce among bowls; top with the warm quinoa, the chickpea mixture and olives. Drizzle with the feta dressing and sprinkle with oregano.