Place 2 tablespoons butter in a microwave-safe bowl; melt in microwave on high, about 30 seconds. Stir buttery round crackers into melted butter.
Place remaining 2 tablespoons butter in a 7x11-inch baking dish. Melt in the preheated oven, 1 to 3 minutes. Remove dish from oven.
Coat both sides of cod in melted butter in the baking dish.
Bake cod in the preheated oven for 10 minutes. Remove from oven; top with lemon juice, wine, and cracker mixture. Place back in oven and bake until fish is opaque and flakes easily with a fork, about 10 more minutes.
Garnish baked cod with parsley and green onion. Serve with lemon wedges.