Can I substitute powdered walnuts for all-purpose flour in a cake?

Sort of. Nut flours are the base of many "flourless" cakes and torts. This one from Joy of Baking is typical, and uses almond flour, which is more common than walnut. Here's one that specifically uses walnut. Note that these are not examples of using nut flours as a substitution for flour, they are recipes developed for nut flours. Both of those recipes start with whole nuts that are ground in a food processor or spice grinder.

Wheat flour behaves very differently, if you want to bake a cake using nut flour in combination with or instead of wheat flour, search for recipes that call for nut flour. Trying to work out an actual substitution would be fraught with peril. Nuts are full of fat and non-gluten protein, wheat flour has starch and gluten forming protein (although a high gluten level is not usually desirable in cakes, that's why cake flour is lower in protein than AP, so that it creates less gluten).

Now I see from the edit to the question that the zucchini/chocolate cake that led to the qustion is actually a kind of cheesecake. Cheesecakes generally don't contain flour, in the case of your recipe the nuts are actually replacing a portion of the cream cheese. In other cheesecakes you might see nuts in the crust or garnish, but I've never seen flour in the filling.