After the dog eats grapes they will usually begin to get sick within 24 to 48 hours after ingesting the grapes.
The way grapes harm and sometimes kills dogs is through kidney failure which happens rapidly as the grapes are toxic to dogs and causes kidney failure in the dog.
The dog may then begin vomiting the grape residue out and may even poop the grape residue out through their intestines as well.
If you think your dog has ingested more than one grape you should take the dog to the vet as soon as possible if it's possible to do so.
If the dog has only eaten one grape then likely the dog will just vomit it out and nothing bad will happen as it usually takes 2 or 5 grapes for the dog to become deathly ill from the grapes and to get kidney failure.
Still it's a good idea to get the dog to the vet to be checked out so that the dog doesn't get kidney failure as a result of the grapes.