Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, whole wheat flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt.
In a large bowl, beat the eggs and granulated sugar together until smooth and slightly thickened.
Add the vegetable oil and vanilla extract to the egg mixture and mix until combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined.
Fold in the grated zucchini, grated apple, and chopped walnuts until evenly distributed.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.