On reading through your linked document, FusionAuth doesn't support this natively. There's no 'sso' endpoint which does what the docs say must be done (checking the signature, creating the new payload, etc...).

You have a couple of options:

file a feature request: https://github.com/fusionauth/fusionauth-issues/issues explaining what you'd like to have done use OIDC for discourse (which should work with FusionAuth out of the box): https://meta.discourse.org/t/openid-connect-authentication-plugin/103632 set up a small proxy server which would receive the SSO request from discourse, present a login screen, and call the FusionAuth Login API to authenticate the user

I'd probably recommend the OIDC route unless there's some reason why it wouldn't work for you.