Place chicken quarters in a 9x13-inch casserole dish. Arrange onion, plantain, figs, and garlic around chicken quarters and season with rosemary, salt, and pepper.
Whisk olive oil and balsamic vinegar together in a bowl; pour over chicken quarters.
Pour water into casserole dish.
Place casserole dish in the preheated oven; reduce temperature to 325 degrees F (165 degrees C).
Bake until no longer pink at the bone and the juices run clear, about 1 hour. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C).