Preheat the oven to 350°F (175°C). Grease and flour a 9×5-inch loaf pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg.
In a large bowl, beat the granulated sugar, light brown sugar, and eggs together until smooth and creamy.
Add the vegetable oil and vanilla extract to the sugar and egg mixture and mix until combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the grated zucchini and grated apple until evenly distributed.
If using, fold in the chopped walnuts.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove the bread from the oven and let it cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing.