Preheat the oven to 350°F (175°C) and grease a 9x5 inch loaf pan with butter or non-stick spray.
In a medium mixing bowl, sift together the all-purpose flour, cocoa powder, baking soda, and salt.
In a separate large bowl, whisk the mashed bananas, melted butter, brown sugar, egg, and vanilla extract until well combined.
Gradually fold the dry ingredients into the wet mixture using a spatula until just combined, ensuring no large pockets of flour remain.
Gently fold in the shredded zucchini and 1/2 cup of the chocolate chips.
Transfer the batter to the prepared loaf pan and smooth the top with a spatula.
Sprinkle the remaining 1/4 cup of chocolate chips evenly over the surface of the batter.
Bake for 55 to 65 minutes, or until a cake tester inserted into the center comes out clean or with only a few moist crumbs.
Cool the bread in the pan for 15 minutes before transferring to a wire rack to cool completely.