Can be found by Dog if you tell him to find something interesting. It can be found as early as the encounter with Dog on the way to Lothering (if you're not a human noble character). At first Dog may not find anything of interest, but when asked again will find it for you.
During the Human Noble origin it is possible to obtain Piece of Wood in your very first encounter with Dog. A dialogue option will allow you to ask Dog if he sees anything interesting just prior to the rat killing quest in Nan's Larder.