Whisk together the lemon juice, vinegar, oregano, garlic and some salt and pepper in a small bowl. Slowly whisk in the oil just until emulsified. Let sit at room temperature while you prepare the salad to allow the flavors to meld.
Rinse the quinoa in a strainer until the water runs clear. Combine the quinoa with salted water and cook about 15 minutes; drain any remaining water that has not been absorbed.
Transfer quinoa to a bowl, fluff with a fork and let it sit for 5 minutes to cool slightly. Add the tomatoes, olives, green onions, cherry peppers, onions, cucumbers and dressing and toss to coat. Cover and refrigerate for at least 1 hour and up to 8 hours before serving. The longer it sits the better the flavor.
Just before serving, transfer to a platter and sprinkle feta cheese on top.