Here is the championship count for the other main conferences:
This is using only teams current conferences, as going back and finding the each team's conference at the time of the championship would be too much. However had I done this the Pacific 10 would still be on top as Arizona's championship and UCLA's 11 championships were won while playing in the AAWU/Pac-8/Pac-10, all of which were name changes of the same conference. Oregon and California won their national championships while playing in the PCC. The Pac-10 is generally considered to be the successor to the PCC although they are not the same organization.
Across all sports, the Pac-12 has more national championships than any other conference.
In College Football, it is difficult to determine which conference has more national championships than another for several reasons. First, teams shift conference affiliations from time to time. For example, Nebraska won three national titles int he 1990s as a member of the Big 12, but Nebraska is now a member of the Big Ten. The Big Ten now claims the titles as having been won by a member institution even though the titles were not won when Nebraska was a member institution.
Additionally, there have often been multiple claimants to the title of "national champion" in any given season. As the NCAA does not award a championship of its own in the Divison I-FBS category, it has been historically left to the polls to decide the "national champion."
Finally, the long history of college football leads to early skewed numbers when relatively few teams participated. Princeton and Yale, no longer part of the top division of college football, claim 44 national championships between the two schools. Add eight from Harvard, six from Penn, three from Cornell and one from Dartmouth would mean the Ivy League far and away has more championships (62) than any other conference. Then again, the Ivy League has not won a "major" college football championship since 1935 (Princeton).
SEC - (10)
ACC - (10)
Pac-10 with 400
Pac 10 with 400
PAC 10, by far
SEC in football
The University of Kentucky. As of 2008, they have appeared in the NCAA tournament 50 times.