Civil registration was not introduced in Austria until after annexation by Germany in 1938. Until then, the churches had the obligation to register vital events on behalf of the state.

You need to visit the web site of the Diocesan Archive in Klagenfurt to find out whether the records of Feistritz an der Gail are still with the parish or at the archive. The archive is willing to make lookups, but as far as I know no searches.