This is a phrase used in baseball meaning there are very few hits in the game by one or both teams. This usually occurs when a pitcher is having a dominating performance. It may also be the case that a team is in a hitting slump or simply does not have many good players. The phrase can be adapted to apply to a single player or group of innings, as in, "Joe's bat has been quiet all season," or, "The team's bats have been quiet the past few innings."
This phrase is considered slang, and is used primarily by fans and broadcasters. There are no official records for tracking quiet bats per se. However, a perfect game and a no-hitter imply that the bats have been quiet if not completely silenced, and those pitching statistics are officially tracked. If an individual player has a quiet bat for an extended period of time, it is known as a "slump."