The answer to this can be found very near to the beginning of Chapter 9. It is on p. 158 in the paperback edition of the book. Diamond does not, however, give any sort of an exact date for when hamsters became pets. Instead, he simply says that it happened in the last couple hundred years. As he says:
Small mammals domesticated as recently as the 19th and 20th centuries include foxes, mink, and chinchillas grown for fur and hamsters kept as pets.
The timing of the domestication of hamsters as pets is really not very relevant to the message of Chapter 9 or of the book as a whole. In Chapter 9, Diamond is trying to argue that there are only a very few large animals that are suitable for domestication. He is arguing that societies that arose in places where such animals lived were lucky compared to those that arose in places where there were no large animals that could be domesticated.
Diamond only mentions hamsters and other small animals in passing. He says that they are not as important for food and they are not at all useful for pulling plows, carrying people, or the like.