Preheat the oven to 350° F. Line a 9x5 loaf pan with parchment paper leaving some overhang for easy removal, and set aside.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large bowl, whisk together the greek yogurt, sugar, eggs, lemon zest, lemon juice, vanilla, and oil until well mixed. Add the dry ingredients to the wet and mix just until there are no streaks of flour left. Be careful not to overmix.
In a small bowl, toss the diced blackberries in 1 tablespoon of flour, and then gently fold them into the batter.
Pour the batter into the loaf pan and bake for 60-75 minutes, or until a toothpick inserted into the middle comes out clean. If the top of your bread is browning too quickly, cover loosely with foil for the last 10 to 15 minutes of baking.
Allow the cake to cool in the pan for about 20 minutes, and then carefully remove to cool completely on a wire rack.