If a URL like path/to/page#this-location?q=word is used the q parameter is ignored. If a URL like path/to/page#this-location&?q=word is used then the query is added to the site search results. The expectation is that any URL with a valid search parameter will work.

It seems Matomo concatenates the querystring and the fragment with an & and then tries to parse the whole string as querystring

On some systems it is not possible to modify the URL to append an & so the site search report does not record these queries.