The murders had occurred overnight, a spokesman for the regional Tirol police said.
The suspected perpetrator had been detained, Austrian press agency APA reported, citing police.
Local media, citing police information, reported the perpetrator was a 25-year-old man who had entered the family home of his former girlfriend and shot five people - the young woman and her new boyfriend, as well as her parents and brother. He later turned himself over to the police and confessed to the murders, media reported.The police spokesman declined comment to Reuters on the local media reports.