Preheat the oven to 350 degrees F. Generously coat a 9- by 5-inch loaf pan with cooking spray.
Spread the walnuts out on a baking sheet and toast in the oven until lightly golden, about 7 minutes. Cool and roughly chop.
Coarsely grate the zucchini into a bowl and squeeze out the excess liquid; you should have about 1 1/2 cups zucchini.
Whisk together the flour, cinnamon, baking powder, baking soda and salt in a medium bowl.
Beat the sugar, oil, apple butter, vanilla and eggs with an electric mixer on medium speed in a large bowl. Add the flour mixture and beat on low speed until just combined, and then on medium speed for 30 seconds. Stir in the zucchini and nuts until evenly combined. Pour the batter into the loaf pan.
Bake until a toothpick inserted in the center comes out clean, 50 to 55 minutes. Cool in the pan on a wire rack for 20 minutes. Remove the bread from the pan and completely cool. Cut into 24 slices and serve at room temperature.