Does anyone know if a Yongnuo trigger YN622C-TX (Canon) will be able to trigger the YN685 (Nikon) Flash unit?

AFAIK, Yongnuo's YN-622 triggering system does not work cross-brand.

See: Can I trigger Yongnuo YN-622N and YN-622C together at the same time?

I have a full kit for my Nikon but now have a Fuji system with seems to run the same mount as Canon.

Fuji's flash hotshoe protocol is not the same as Canon's. Just because the contacts/pins are in the same place doesn't mean the electronic signalling across those contacts/pins is the same. Ditto micro four-thirds. Or Pentax/Nikon. Fuji TTL/HSS is not supported by Yongnuo; they only support Canon and Nikon TTL/HSS.

You might be able to put the YN-622C-TX into a single-pin or 603 mode and use it as a manual-only trigger. But you won't have TTL/HSS from a Fuji hotshoe, and I wouldn't count on power control.

Also, if your Fuji body isn't an X-T1, X-T2, X-T20, or X-Pro2 (or newer body), you won't have HSS anyway, since Fuji only added that as a feature in 2016, and those are the only four bodies that can do it.

So hoping I would only need to buy a trigger and not a whole new system.

If you want TTL/HSS with Fuji, you have to get a whole new system that supports that. You may want to look at triggering systems like Godox X/Flashpoint R2, Nissin Air, or Cactus V6II, which let you share the lights across brands with different system-compatible transmitters. The Profoto Air system does this as well, but might be prohibitively expensive. All of these systems support cross-brand TTL/HSS for Nikon, Canon, Sony, Fuji, and Four-Thirds.