Preheat oven to 350°F.
In a bowl, whisk together the flour, baking powder, baking soda, and salt.
In the bowl of a standing mixer, or with an electric mixer, cream together the butter and sugar until light and fluffy, about 3 minutes. Beat in the egg and vanilla.
Gradually stir the flour mixture into the butter mixture, until just combined. Fold in the zucchini, coconut and walnuts.
Pour batter into a greased 8-inch square cake pan.
Bake for 30-40 minutes, or until a wooden pick inserted in the center comes out clean.
Serve as is or allow to cool completely and then frost.
To make the cream cheese frosting: combine ingredients in a bowl and stir to combine well.