Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry for this concern for your dogs. Potatoes, particularly green ones, contain solanine which can cause gastrointestinal irritation but if enough is ingested, it can also cause muscle tremors and cardiac involvement too.
As the the plant ages, the toxin is destroyed so how likely you are to see side effects, depends on how much was ingested and how mature the potato was. I wouldn't expect ingestion of one potato by one dog to be of tremendous concern, though; usually it's when dogs eats pounds of them that we worry.
It might cause diarrhea in 12-24 hours but that's about all I'd expect to see. If looser stools should develop, then a bland diet of chicken and rice should help return the stools to normal.
If we knew for certain that only one of the dogs ate the potato and/or many of them were eaten, then I might suggest that you induce vomiting if the ingestion took place less than 1 hour ago. But, under the circumstances, I wouldn't think this would be warranted.
I hope this helps to provide reassurance about them. Deb