Yes, there are a few ways to do it. You need to have logged into the Fossil app with an email/password, it didn't seem to work with "login with Google."
I used a python script that I ran on my computer that got the private key from the Fossil servers.
https://gadgetbridge.org/basics/pairing/fossil-server/
NOTE: You need to add the private key before pairing with GadgetBridge, if you add it after then it won't work.
Short skirt / Long jacket?