Do you think we can use [winning the question contest] on our cover letters?
No. Very few of my colleagues even know this blog exists, so they would have no idea what you were talking about. Besides, this is not a legit writing contest.
I know you were joking, but here's what you can use--legitimate contests that offer real prizes of cash or publication, like Writer's Digest contests. Being published in an anthology does not count.
How to tell if the contest is legit? Check Preditors and Editors.