Whisk white wine, oil, oregano, pesto, lemon zest and juice, garlic, salt, and pepper together for marinade in a small bowl.
Put the chicken leg quarters in a resealable plastic bag and pour marinade over chicken. Seal the bag and shake to completely cover. Place in the refrigerator to marinate for at least 2 hours, turning over after 1 hour to ensure chicken quarters marinate evenly.
Preheat the oven to 425 degrees F (220 degrees C). Bring chicken to room temperature while the oven heats.
Drain marinade from chicken and pat dry; place in a baking dish.
Roast in the preheated oven for 20 minutes. Turn chicken and cook until no longer pink at the bone and the juices run clear, about 20 minutes more. An instant-read thermometer inserted near the bone of a quarter should read 165 degrees F (74 degrees C).
Meanwhile, during the last 20 minutes of baking the chicken, bring goat cheese to room temperature and make sauce.
Heat butter in a skillet over medium heat. Saute onion and garlic in the hot pan until soft and translucent, about 5 minutes. Pour in white wine and lemon juice. Add sun-dried tomatoes and pesto and season with salt and pepper. Reduce heat and let simmer until slightly thickened, about 15 minutes.
Remove chicken from the oven and top each chicken quarter with 2 slices of goat cheese, 1 on each thigh and 1 on each leg. Cover with equal amounts of sauce and serve immediately.