Short answer
Yes, that's a common reaction to rich, fatty goose meat. It can easily upset a dog's stomach, leading to loose stools or diarrhea. Offer plenty of water and skip the next meal to let their system settle. If the diarrhea continues beyond a day or they seem lethargic, give your regular vet a call.
Goose — Food safety facts
Safe to share · Dogs · full breakdown on the linked food page
Cooked goose meat is safe, but it is fatty — keep portions small.
The full verdict, portion guidance and serving notes are collected on the Goose food facts page.