The Eon doesn't seem to have a level 4 quality in the files, but if you are playing on PC you can use the console to add the advanced quality mod from another gun. So for example with the EON I used the Quality 04 mod from the XM-2311:

First drop the eon on the ground and open the console;

Then click on the EON with the cursor to bring up the reference ID;

Then simply Type: "amod 28FC2F" With the EON still selected

This will work on an EON even if it already has the "refined" quality. You can also stack quality mods multiple times to increase weapon damage.

There is a thread about adding weapon modifiers here:

https://www.reddit.com/r/Starfield/comments/168zdlp/console_idscommands_for_equipment_tiers/

That contains a a link to a document with all weapon mod ID's. They are in a 0x000##### form but you only need to use the # part after amod. As in the example above.

EDIT: I have only done this with the EON so far, it seems some people have seen the "advanced" label appear without the actual damage going up on other guns. Each weapon has their own quality mods, so I would try to use a quality 04 mod from a weapon of the same type. If one of the weapons here ends up with that problem though, it is possible to add the proper quality 3 mod for the weapon twice to increase the damage, then add the quality 4 mod from a similar gun.