November 24, 2007. That was the last time the UCLA Football team shutout an opponent as the Ben Olson-led Bruins shocked the #9 Oregon Ducks 16-0. Not to say that shutouts are a direct representative of a good defense but they do show a correlation. Limiting your opponents to not scoring does increase your chances of victory.
Since that game against the Ducks, there have only been five other games in which UCLA has held their opponents to 7 points or under: 2009 and 2008 vs Washington State, 2008 vs Washington, 2012 vs Houston and last season against Arizona.