Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a large mixing bowl, whisk together the flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
Add the melted butter and sugar to the dry ingredients and stir until combined.
Add the eggs, one at a time, mixing well after each addition.
Stir in the vanilla extract and Greek yogurt until the batter is smooth.
Fold in the diced apples and chopped walnuts (if using) 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 let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.