R. Tyler Croy I think this one could be made clearer. I think I see what you mean, but ideally evergreen tasks should be doable not only by us knowing well both Jenkins and Evergreen context.
IIUC, this is about the so-called Administrative Monitor displayed under /configure when a new version of Jenkins is available .
If so, there's already some food for thought in https://groups.google.com/forum/#!msg/jenkinsci-users/TSY2MhkTNJ4/8Qc9yMFGBgAJ
More specfically: https://groups.google.com/d/msg/jenkinsci-users/TSY2MhkTNJ4/mLgot1oKAgAJ
Go to Manage Jenkins> Configure System> Administrative monitors configuration and uncheck Jenkins Update Notification box
and save the settings and refresh jenkins page. Now you should not see the jenkins new version update notification.
I think the right way inside Evergreen to achieve this is to find a way to "check" this checkbox (i.e. disable this administrative monitor) using JCasC.
Seems like the code inside Jenkins is https://github.com/jenkinsci/jenkins/blob/b057abc7b84295969b894d06011e250842c6d09c/core/src/main/java/hudson/model/UpdateCenter.java#L981-L999