The 28-year-old Rogers, who was outstanding in the 40-10 win over Parramatta in round 10, is the last man standing in the Broncos No.7 roster.
Rogers, who plays for Burleigh in the Queensland Cup, is on the club's development list and outside the top 30.
Brisbane will be short on middle and edge forwards for the Dragons clash due to prop Payne Haas and lock Pat Carrigan being on State of Origin duty.
The Broncos have just added prop Va'a Semu to their development list for 2024 and signed the 194cm and 110kg giant to a further three-year deal until the end of 2027.
Semu has played 13 games in the Queensland Cup this season for Souths-Logan, mostly off the bench, and has an outstanding defensive record.
AAP understand Semu is being considered for an NRL debut from outside the top 30, given that forwards Ben Te Kura, Jaiyden Hunt, Fletcher Baker and Jack Gosiewski are all unavailable due to injury.
Madden is expected to miss at least a month after sustaining his injury in the 43rd minute of the 14-6 loss to Penrith on Friday night.
The Broncos are already missing captain and first-choice half Adam Reynolds with a biceps tear until round 22 or 23.
Teenage half Coby Black, who is yet to play NRL, has a broken thumb and is also unavailable.
"Josh Rogers has the bye this week (for Burleigh) so he will be right to go against the Dragons," Walters said.
"He was outstanding (against the Eels). It is unfortunate for Jock but very good for him to get another crack at this level."
The Broncos do have hooker Billy Walters and utility Tristan Sailor who can play in the halves but Walters will stay at No.9 and Sailor is likely to play fullback against the Dragons and replace fullback Reece Walsh who will be on State of Origin duty.
Madden, who has filled in for Reynolds, spoke after the Panthers match about his injury.
"I am very disappointed obviously but it is out of my control," he said. "I have torn my hammie but we will wait and see how the scan goes and I will focus on my recovery.
"(Rogers) will do a great job when he comes in. He is a good kicker and can manage a game really well."
The Broncos have now lost five games in a row and are in danger of missing the top eight.
Next Saturday night they will lose at least Walsh, Haas and Carrigan to Origin.
Centre Selwyn Cobbo is also likely to earn a Maroons recall as well.
"We are going to have to go outside the top 30 now. We have four possibly in Origin but we will get enough jerseys and be out there to take on the Dragons," Walters said.
Centre Kotoni Staggs will return next week after missing the Panthers clash to attend a close friend's funeral. Second-rower Jordan Riki will also be back from suspension.
