I think it would be best if docs.julialang.org adhered to the OpenSearch standard to advertize itself as a search engine. Most mainstream browsers seem to understand this specification (which is used by Discourse, BTW).
In the meantime, I’ve put together a small webpage to provide OpenSearch specifications for the Julia Reference manual: http://ffevotte.github.io/juliaOpenSearch/index.html
I you go to the page, your browser should propose you to add a new search engine. In Firefox, it looks like this (in the top-right part of the window):
Once the search engine is added. You can affect it a shortcut. Again in the example of Firefox, this is performed by clicking the “Change Search Settings” button visible on the screenshot above, or visiting the preferences menu (about:preferences#search)
PS: the OpenSearch document for the Julia documentation is here, in case you’d want to inspect it beforehand: http://ffevotte.github.io/juliaOpenSearch/julia.xml