http://jenkins-url/api or core/src/main/resources/jenkins/model/Jenkins/_api.jelly in the code have about the http://jenkins-url/quietDown API:

Jenkins will enter into the "quiet down" mode by sending a POST request with optional <code>reason</code> query parameter to <a href="../quietDown">this URL</a>

But the reason parameter seems to be ignored. It seems the parameter one needs to supply to indicate the reason is message instead.

Looks like it initially was reason but was changed to message later, and the documentation not updated.