The Reds have endured a mixed start to their Premier League campaign and were knocked out of the Capital One Cup last week at the hands of Rodgers's former employers Swansea City.
Despite some supporters voicing their concern about the direction that the club is heading in, McDermott has urged the Liverpool faithful to get behind the boss and the club.
"[Questions are going to be asked] when you're not getting results," McDermott told talkSPORT. "They've played some good stuff like they did last year under Kenny [Dalglish] but they weren't getting results at home that the performances deserved and it seems to be happening again.
"I'm disappointed they went out on Tuesday because they were the holders of the Capital One Cup. I would have played my best team, that's my opinion.
"I think Brendan Rodgers is going to be a very good manager of Liverpool. I don't think they should have got rid of Kenny but they did and now Brendan's in and I suggest every Liverpool supporter gets behind them. He is a good manager and I'm sure he's going to be a better manager but it's a difficult one to take when you get beaten at home to Swansea when really you should be in the quarter-finals."
Liverpool travel to Chelsea in their next match on Sunday.