Below is the list of undefeated teams since 2000 (I didn't look earlier). No quarterback injuries.
2000 Oklahoma - Josh Heupel
2001 Miami - Ken Dorsey
2002 Ohio State - Craig Krenzel
2004 USC - Matt Leinart
2004 Auburn - Jason Campbell
2004 Utah - Alex Smith
2005 Texas - Vince Young
2006 Boise State - Jared Zabransky
2008 Utah - Brian Johnson
2009 Boise State - Kellen Moore
2009 Alabama - Greg McElroy
2010 TCU - Andy Dalton
2011 Auburn - Cam Newton
2012 Ohio State - Braxton Miller
2013 Florida State - Jameis Winston