Johnson said it was in the best interests of the club if he remained as the club continued its search for a replacement for coach Michael Voss who was sacked last week.
Former triple premiership-winning Lions coach Matthews has joined forces with Brisbane directors Paul Williams and Mick Power in a rival ticket that will challenge Johnson.
While Johnson's fate initially looked set to be decided at an emergency meeting, Matthews' ticket will not get a chance to topple Johnson until the club's AGM in December.
Johnson refused to stand down when challenged by Williams and Power at a Lions board meeting on Wednesday morning.
"This came as a surprise," Johnson said of the attempted coup.
It is understood Matthews, Williams and Power have the support of Lions powerbrokers who have lost faith in Johnson since Voss' sacking and want to launch a fresh push to secure former Sydney premiership-winning coach Paul Roos.
A successful coup could result in Matthews being ushered into a football director's role at the Lions.
Matthews said he had knocked back several approaches to return to the Lions since sensationally ending his coaching stint in 2008.
However, he changed his tune and decided to join a ticket when convinced the club's best future was under a Power-Williams led board.
"My jump over the cliff was when Paul and Mick said they were going to run for the chairmanship and deputy chairmanship," he told The Courier-Mail website.
"I know these people well and believe in what they are doing.
"I'm on because of them and I believe I can help."