Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan and set aside.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg.
In a large bowl, beat the granulated sugar, light brown sugar, and eggs until well combined.
Add the vegetable oil and vanilla extract to the sugar and egg mixture and mix until smooth.
Gradually add the dry ingredients to the wet ingredients, stirring until just combined.
Fold in the grated zucchini, chopped apple, and chopped walnuts (if using) until evenly distributed.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
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 and let it cool completely before slicing.