He will officially become a Bees player on July 1.
Bonham was thrust into the spotlight at the end of last season when he made his Watford debut as a substitute against Leeds following injuries to Manuel Almunia and Jonathan Bond.
He was at fault for Leeds' winning goal as Watford's automatic promotion hopes came to an end.
But Brentford feel they have snapped up a talented young keeper, with sporting director Mark Warburton telling the club website: "We are delighted Jack has agreed to sign.
"He spent two weeks with us on trial at the back end of last season and did well.
"Jack has had an excellent Academy education with Watford, developing into a fine goalkeeper, with an international pedigree.
"He has enormous potential and will improve by being within the goalkeeping unit and working with coach Simon Royce."