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.
Make lemon glaze
In a bowl, whisk powdered sugar and lemon juice together until smooth and clumps remain. If your glaze is too thick, add a teaspoon of lemon juice until you reach the desired consistency. If it's too thin, add powdered sugar.
Once your loaf is cooled completely, pour glaze over the top and spread around. Allow the glaze to harden for about 20 minutes before cutting into the bread.
Video
Notes
Make sure to line your pan with parchment paper and leave a little overhang, it makes it so much easier to remove after baking.
If you want a stronger lemon flavor in your bread, you can add an extra tablespoon of lemon zest (not juice).
Cut your blackberries in half or quarters before tossing them with the flour.
I always recommend spooning and leveling your flour when baking. Or for more accurate measurements, use a kitchen scale and weigh your ingredients.