Preheat the oven to 350°F (175°C). Grease and flour a 9×5-inch loaf pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, beat the granulated sugar, 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 fully incorporated.
Fold the dry ingredients into the wet ingredients gently until just combined; do not overmix.
Fold in the grated zucchini, grated apple, and chopped walnuts (if using) until evenly distributed throughout the batter.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake in the preheated oven for 45 to 50 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 15 minutes.
Transfer the cake to a wire rack and let it cool completely before slicing and serving.