Other than the Duo application's own logging, not really.
We are working on implementing Desktop SSO as part of Duo Single Sign-On. If that might be of interest to you please reach out to your Duo customer success manager or Duo account manager. If you do not work with a Duo contact now, you can contact Duo Support (https://duo.com/support) to be added to any existing feature requests for Windows desktop SSO, and you may possibly be contacted by a Duo product manager to discuss your use case.
Duo, not DUO.