Is there a way to update the maps on my fēnix 6 without access to a Mac or Windows machine?

You can't (as far as I know) upgrade the official Garmin maps without Garmin Express, but you certainly can use unofficial sources based on OpenStreetMap which are usually better and updated more regularly (and free if you want maps outside your region)

Example:

https://garmin.opentopomap.org/

Just copy the .img files to /GARMIN

Does Garmin Express not work with WINE?

I tried a walkthrough I found online a while back, but it errorred out. Also tried using Gnome Boxes + W10 back when I was on Gnome desktop. Couldn't see the device from W10.

I might try this. Saw an old DC Rainmaker post about this. Have to see how the maps work on a watch screen. Probably worth a shot though. From experience of comparing Open Street Map to Ordnance Survey & Garmin, Garmin seems to have the worst public footpath maps around Oxford's countryside.

However, I'd still like to be able to update the Garmin maps I have. They came with the device and whatnot.

OceanMantaray said:

comparing Open Street Map to Ordnance Survey & Garmin

The Garmin maps installed are "Topo Active" maps, which are also based on Openstreetmap.

OceanMantaray said:

I'd still like to be able to update the Garmin maps

Then you need a Windows or MAC system.

bluefish said:

The Garmin maps installed are "Topo Active" maps, which are also based on Openstreetmap.

While this may be the case, the maps produced by Garmin still have less public footpaths marked that Open Street Map or Ordnance Survey. I do not know why this is the case. Simply an observation.

I agree, even using the Garmin Explore app on my phone there are lots of paths in my area that are not shown on the map, but are shown on other apps such as Komoot, Suunto, mapy.CZ etc!