Judge, who can play as a midfielder or as a winger, suffered a season-ending injury against Ipswich Town away from home on Saturday afternoon.
The 27-year-old broke his right leg during the Championship game at Portman Road and had to leave the pitch on a stretcher.
According to The Guardian, the Republic of Ireland international will miss the rest of the season and will also sit of the Euro 2016 finals in the summer.
Dallas, who joined Leeds from Brentford in the summer of 2015, has taken to Twitter to send a heartfelt message to his former teammate.
Devastated for @10judgey Wishing you a speedy recovery pal.— stuart dallas (@dallas_stuart) April 9, 2016
Brentford won 3-1 against Ipswich and are now 13th in the Championship table.
Judge has been superb for the Bees this season, and has scored 14 goals and created 88 chances in 37 Championship appearances.
The former Blackburn Rovers man has a pass accuracy of 84% and a shot accuracy of 52%, and has won 10 headers, 31 dribbles and 49 tackles.