Blackberries can fit as an occasional add-on, not a daily extra.
Quick call
Yes, in moderation. Too many berries can cause loose stool.
What matters most
Offer washed fresh berries, plain.
When to call your vet
If your dog has vomiting, diarrhea, unusual fatigue, or belly pain after eating blackberries, call your vet the same day.
Source check: PetMD lists blackberries as an occasional fruit treat for dogs. Source