You can find the names of the two feuding families very early in the short story “The Interlopers.” The two families are the von Gradwitz family and the Znaeym family.
These two families have been feuding with one another for a few generations now. The two protagonists in the story are the current heads of the families. They are continuing a feud that began in the days of their grandfathers. In those days, their grandfathers engaged in a lawsuit with one another regarding the control of a certain parcel of land. Both men claimed that it was theirs, but the court awarded the land to the von Gradwitz family. Ever since, the two families have been feuding, though the feud has “come to a head” in recent times.
The answer, then, is that the two families are the von Gradwitz family and the Znaeym family.