Police said Constable Gellatly's family have been informed of his death and officers from an independent force, Lothian and Borders Police, will oversee the investigation.
Assistant Chief Constable Bernard Higgins said: "It is with great sadness that I confirm a serving
Strathclyde Police officer died today in an incident where a firearm was discharged at Baird Street Police Office.
"Our thoughts are with the officer's family and friends at this difficult time."
Brian Docherty, chairman of the Scottish Police Federation, said: "It is too early to speculate as to the circumstances surrounding the tragic death and our thoughts turn immediately to his family who have lost their loved one and colleagues who have lost a friend."
More top news
Erin Sandilands was told to apply more makeup, wear a skirt or put her hair down because "it's better for the punters".
Theresa May will make her pitch to lead the Conservative party today, 90 minutes before long-time favourite Boris Johnson.
Cool temperatures and breezy conditions with a top temperature of 20 Celsius (68 F).