Argentina captain Montoya was voted Player of the Year by team-mates and also by supporters, while full-back Steward was similarly recognised in the Young Player of the Year categories.
Montoya joined Tigers in at the start of 2021 and has made 19 appearances this season.
Accepting the trophies as Players’ Player and Supporters’ Player of the Year, he said: “I am very proud and honoured to do this and I thank all of my team-mates and the supporters.”
Steward also picked up two awards last week at the Rugby Players’ Association event, Including as England Men’s Player of the Year after making his international debut last summer.
Freddie Burns became the first-ever two-time winner of the Tigers Newcomer of the Year award after topping voting among supporters.
Burns previously picked up the same award in 2015 and returned for a second spell at the club last summer, making 23 appearances so far this campaign and passing 100 games for the club during his last outing, at Newcastle Falcons 10 days ago.
“When I was in Japan I told my agent I would come back to the Premiership only if it was Leicester and I was delighted to come back here,” he said.
The Topps Tiles try-Umph of the Season was voted for exclusively by guests at the Annual Awards at Mattioli Woods Welford Road for the awards event on Monday evening from a shortlist of the monthly winners throughout the season.
Chris Ashton, for his third score in the home win over Bristol Bears last month, came out on top of the voting and was presented with his award.
The Outstanding Service award went to Tom Youngs who last month announced his retirement as a player and earned an emotional reception from supporters as he led out the team for the win over Bristol Bears at Mattioli Woods Welford Road.
Winner of the Leicester Tigers Foundation awards was Matt Johnson.
The evening was organised in support of the Tigers Foundation and its work within the community, including with the Leicester Tigers Wheelchair Rugby Team.
Special thanks also goes to event sponsor PPL PRS.
Tigers Awards winners 2021/22:
Supporters’ Player of the Year, sponsored by IMA Architects: Julián Montoya
Players’ Player of the Year, sponsored by G&G Travel : Julián Montoya
Supporters’ Young Player of the Year, sponsored by Wellgosh: Freddie Steward
Players’ Young Player of the Year sponsored by STIHL: Freddie Steward
Topps Tiles Try-Umph of the Season: Chris Ashton
Outstanding Service: Tom Youngs
Newcomer of the Year, sponsored by Fidelity Group: Freddie Burns
Leicester Tigers Foundation: Matt Johnson