The actor is currently working in the city - and marked the special occasion with friends at Rose Street Garden on George Street.
The venue shared footage online of the actor’s 37th birthday bash.
Staff members held out placards spelling out his name as they served cocktails at his table.
The footage drew lots of comments, and over 1600 likes on the Rose Street Garden’s Instagram page.
Greenock-born Compston stars as DI Steve Arnott in the hit series which drew a record audience or 12.8million for the finale of series six last week - one of the most talked about television moments of 2021 so far.
The BBC said it was the most watched episode of any drama since modern records began in 2002 - excluding soap operas.
The series finale took 56.2% of the UK's TV audience.