Preheat oven to 350 F. In a stand mixer, or by hand if your feeling strong, cream together butter and sugar until light and fluffy. Beat in eggs, lemon juice and zest.
In a separate medium bowl, whisk together, baking powder, salt and flour. Add alternately with the Greek yogurt to the wet ingredients. Mix until just combined. Gently fold in blackberries.
Bake at 350 F. for 55-60 minutes until an inserted toothpick comes out with a few moist crumbs clinging to it. Let cool in pan for 15 minutes before inverting onto wire rack. Let cool completely.
To make the glaze, combine blackberries, water and sugar in a small saucepan on the stove, Simmer for about 5 minutes over medium heat until the blackberries are very soft.
Strain through a fine mesh strainer to remove the seeds. Beat in powdered sugar until a glaze forms. If it is too stiff and a drop or two of water at a time. Drizzle over cooled cake.