John's dog. or One of John's dogs.

You can say “a dog of John’s” but this is super uncommon. If you said this in the United States, it would sound funny to most people. Instead, say “one of John’s dogs.” “A dog of John” is not correct.

If you are talking about one dog out of two or more then 'a dog of John's died last week' and 'one of John's dogs died last week' both have the same meaning.