The 31-year-old former Republic of Ireland international had an arthroscopic procedure on Wednesday on his right knee, which had troubled him during the latter stages of the season.
"The procedure was to correct an ongoing condition that had developed over a long period of time and had been restricting his training and matches towards the end of the season," Brentford's head of medical Neil Greig told the club's official website.
"The surgery was delayed following the end of the season in order to allow the knee sufficient time to settle down from the demands of the season.
"This will hopefully make the recovery process a smoother journey for Jonathan.
"Following a period of rest and recovery Jonathan will begin a graduated rehabilitation programme.
"The timescales of this will be easier to estimate following the initial period of rest, as we see how the knee responds post-surgery."