Yes blackberries can cause red stools which is normal.
Other foods such as red berries, strawberries, strawberry milk, beets etc can all make your poop or stool turn red.
So if you've been eating blackberries or other berries or any other foods that are pink or red in color then your stool can become red.
The red stool or red poop usually clears up on it's own after you stop eating the red foods or drinking red drinks.
However if the red poop or red stools continue then it could be something wrong inside your bowels or colon.
So get yourself checked out just to be safe if the red stools or red poop continues.
But eating foods that are red or pink or other colors can naturally and normally change the color of your poop.