Preheat the oven to 350°F (175°C). Lightly grease a 9×5-inch loaf pan and set aside.
In a medium bowl, whisk together the all-purpose flour, whole wheat flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, combine the mashed bananas, shredded zucchini, eggs, granulated sugar, brown sugar, vegetable oil, and vanilla extract. Mix until smooth.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake in the preheated oven for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove the bread from the oven and allow it to cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing and serving.