Mix peaches, blackberries, 1/4 cup sugar, cinnamon, and lemon juice in a bowl.
Pour butter into a 9×13-inch glass baking dish and evenly coat the bottom and sides. Whisk flour, 1 cup sugar, baking powder, and salt in a bowl; mix in milk, stirring just to moisten dry ingredients. Pour batter into prepared baking dish and distribute fruit mixture evenly over the batter.
Bake in the preheated oven until cobbler is golden brown, about 45 minutes.