Breeding dogs? There are two dogs in the painting but this is not their hobby. One dog may be involved in the family's hobby though.
It's not growing fruit either. There are some apples in the picture but they represent something else.
The correct answer is hunting. There are three clues; the hawk, the vicious-looking hunting dog behind Sir Thomas' chair and the spurs on the table. These are all used in hunting. This was probably the hobby of the father and maybe the son.