Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt. Set aside.
In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time to the butter mixture, beating well after each addition.
Mix in the vanilla extract.
Add the dry ingredients to the wet ingredients in three additions, alternating with the sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Fold the diced apples gently into the batter.
Pour the batter into the prepared baking pan and spread evenly.
Bake in the preheated oven for 35 to 40 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 on a wire rack for 15 minutes.
Slice into 8 equal pieces and serve warm or at room temperature.