no online-reseach is available.
civilrecords started in Austria in 1938, before to research at the parish-churches. you can ask at diocesan-archive in Graz for your ancestors: http://www.katholische-kirche-steiermark.at/dioezese/ordinar...

after 1938 > civil-records > Standesamt (registry-office) Graz:
http://www.graz.at/cms/beitrag/10017599/313159/