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, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat the granulated sugar and eggs together until pale and fluffy, about 3 minutes.
Add the Greek yogurt, vegetable oil, and vanilla extract to the egg mixture and mix until smooth and well combined.
Gradually add the dry ingredients to the wet ingredients, folding gently until just combined. Do not overmix.
Fold in the finely chopped pistachios and lemon zest evenly into the batter.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted in 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 and let it cool completely before slicing and serving.