The incident happened this morning under the jurisdiction of Buddha Colony police station, the police said.
A trainee constable, in a fit of rage, opened fire from his service weapon during an argument with his brother-in-law, the Additional Superintendent of Police (ASP) of Patna Central, D Amarkesh, said.
When other policemen tried to stop him, he opened fire at them also, the ASP said.
He said, the constable's brother-in-law was rushed to a hospital where he succumbed to injuries.
The constable was overpowered by other policemen on the spot, he added.
The trainee constable has been arrested and an investigation has been initiated, the ASP said.