LDStorymakers recently had a First Chapter contest for conference attendees and Irreantum has a contest as well. If you enter one of these contests, or another one, and place in it, would it help to mention that in your cover letter?
It won't hurt. These are two contests that LDS publishers would recognize. I've read winning submissions from both of them and they are usually a step above the average slush pile submission.
However, I've seen lots of first chapter winners that fall apart in chapter four.