I don’t know if this is something @manton is planning to add support for or not. But in any case, if you’re okay with rolling your own solution and depend on client-side JavaScript, it’s definitely doable.
A password-protected posts and pages plug-in for Micro.blog has been on my looong some time/maybe-list for a while. Unfortunately, my days have too few hours in them. 
If you want to research this yourself, “password protected static site” or “password protected Hugo posts” are good phrases to start you off. StatiCrypt is one solution.