Barbillus is a wealthy Alexandrian who was a friend of Caecilius prior to Caecilius' death in the eruption of Mount Vesuvius. Barbillus is first introduced in the second book, when Quintus recounts the events of his stay in Alexandria to Cogidubnus.
He gives Quintus a place to stay for free for the time Quintus is in Alexandria. He gets stabbed by his own slave right in the umerum while hunting for crocodiles.
His wife died and his son is off somewhere. He's pretty lonely and sad but hides it well.
He also trusts astrologers a bit too much.