Mix together feta, green olives, black olives, and lemon juice in a small bowl. Season with salt and pepper. Place in the refrigerator until ready to serve.

Combine ground turkey, breadcrumbs, chopped onion, egg, feta, lemon zest, dill, oregano, garlic, and smoked paprika in a bowl. Shape mixture into four 1/4-pound patties.

Preheat an outdoor grill to medium-high heat and lightly oil the grate.

Cook the turkey burgers on the preheated grill until no longer pink in the center and the juices run clear, 7 to 10 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).