Preheat the oven to 350 degrees F. Grease and flour a 9 by 5-inch loaf pan, or line it with parchment paper.
Whisk the egg, brown sugar, oil, granulated sugar, sour cream, and vanilla in a large bowl until smooth.
Add the flour, baking powder, baking soda, and salt. Stir just until combined.
Fold in the grated zucchini gently.
Toss the blueberries with the remaining 1/4 cup flour in a separate bowl.
Fold the coated blueberries and any loose flour into the batter without overmixing.
Transfer the batter to the prepared pan and smooth the top.
Bake for 55 to 60 minutes, until golden and a toothpick inserted in the center comes out clean or with a few moist crumbs.
Cool in the pan for 15 minutes, then transfer to a wire rack and cool completely before slicing.