A comparatively long time,reference 1 though long compared to what is less clear.
Dogs who live into their teens are considered old dogs; almost none make it to 20 years. So perhaps a “dog's age�? is a length of time of less than 20 years. Or it may simply be an interval of time between events that is longer than the expected interval.
In an Oct 24, 1993 New York Times Book Review article discussing crime novels, Marilyn Stasio complained that “Peter Lovesey hasn't written a Sergeant Cribb mystery in a dog's age;�? actually it had been about fifteen years.