It's a very good question - initial gut feeling was no but very close.
Brady, Montana, Elway, Manning are the Rushmore, I think it's tough to make a case against that. So that is 6 spots left containing names such as (in no order) Brees, Marino, Unitas, Staubach, Rodgers, Young, Bradshaw (the Bill Russell factor - he still did win 4 rings), Graham, Bart Starr, Favre and Roethlisberger.
I think it then really depends on your criteria for greatness. Here are the selling points for Ben.
2 rings (still more than Rodgers, Brees, Favre) and played in one more. Truly a man of steel - 19 years and NEVER had a losing record in a season with only a handful of missed games. Led league in passing multiple times and has secretly amassed awesome numbers - 5th in wins, 5th in yards, 5th in playoff yards, 8th in passing TD's. On field could absorb a crazy amount of hit's and maybe was the best QB in history for staying on his feet and extending plays - something rarely talked about.
So honestly, despite that gut feeling I think he is right there at the cusp, I wouldn't be aggrieved if he was anywhere from 9-12 to be honest and the numbers and memories of him would support that. I will also say this, of all the QB's in the modern era there is only one other QB I would want more with the ball in the final 2min needing a game winning drive in a playoff game and that's Brady. I would take BR over Brees, Rodgers, Manning - all of them except TB12