I haven't tried it, but it looks like there is some plans to tweak Moodle to take advantage of some of it's features: https://tracker.moodle.org/browse/MDL-52751
As far as I understand it, because it's fully backwards compatible, you could turn on HTTP/2 at the web server level (Apache or Nginx or whatever) and things should continue working as they normally do. The modern browsers that are capable of HTTP/2 will just continue running as before and if most of your users are using an HTTP/2 capable browser, you just start optimizing for it (in Moodle code or other) rather than the previous version. For example, Wordpress has plugins you can add that may make make the user experience a bit faster by pre-loading scripts/styles: https://wordpress.org/plugins/http2-server-push/
It won't give a free speedup though I think, the website has to be optimized for it's use before it helps.. but again, I haven't tested it.
Also, it's not just a Google thing. Google started the initiative but it's been standardised over 3 years ago now by the IETF. All recent browsers support it, other than Internet Explorer 11 on Windows 7. See https://caniuse.com/#feat=http2
For those of us that only serve up Moodle, I think there would be no benefit right now until wee see Moodle's code take advantage of it, but I am curious also if anyone has done tests to see if the multiplexing built into HTTP/2 speeds things up in Moodle without any code tweaks.