Preheat the oven to 350 degrees. Grease a 9 x 5 inch loaf pan and line it with parchment paper.
In a large bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a separate large bowl with a handheld or stand mixer, beat together the butter and sugar for 2-3 minutes on high.
Add one egg at a time to the butter and sugar. Mix on low until just combined.
On low mix in the yogurt, vanilla, lemon zest, and juice at the same time.
Add in about half of the dry ingredient mixture, mix on low until mostly combined.
Add in the rest of the dry ingredients and mix on low until mostly combined.
Add the blackberries.
Gently fold in the blackberries with a large spatula, scraping any flour stuck to the edge of the bowl.
Pour the batter into the prepared loaf pan.
Smooth the top of the batter into an even layer.
Bake for 55-60 minutes, covering loosely with foil around the 35 minute mark. This will prevent the top from browning too much. Place on a wire rack.
In a small bowl, combine the butter and lemon juice.
Add the powdered sugar.
Mix together until a thick glaze has formed.
Spread the glaze on the loaf while it is still warm. Cool completely before slicing.