I wanted to leave a comment here because I came across this post looking for a this answer, and it may help someone else. (So this is probably only helpful to you as a clue, rather than as a complete answer to your question -- I'm not sure what types of queries you're trying to save.)
I was trying to do something similar with saving the URL of an advanced search to my navigation. My goal was to display results as Memberships (rather than the default contacts) and filtering by contacts in a certain group. (backstory: I needed to display as Memberships so I could export a file [using a saved field mapping] for printing membership cards, and I couldn't access membership info when exporting the contact only.)
This is the url that ended up working (not using qfKey! -- it works in different sessions):
(After clicking the link, it will append the qfKey to the url, but whatever.)
I'll try to break it down, but I don't necessarily know the full extent of these options.
component_mode=[#] is the "display results as" in the search criteria. 5 is Memberships, so that's what I'm using. Here is the inspect elements from my search form (caveat: I have some components disabled; this may change in future versions; etc. so... double check this):
<select class="crm-select2 crm-form-select" name="component_mode" id="component_mode" tabindex="-1" title="Display Results As" style="display: none;">
<option value="5" selected="selected">Memberships</option>
<option value="7">Related Contacts</option>
gid=[#] is the group id that I'm filtering by. I'm not sure what that looks like if you're tying to filter with multiple groups.
force=1 seems to display results, rather than the search criteria
For other search parameters, you basically just have to dig around in the html using inspect element, and find the name/id of the filter you want to use, and then append that to the end of the url. (As far as I can tell, this works for other filters, but obviously my testing was not extensive.)