The official birthday of HM The King takes place each year on a Saturday in early June.
The exact date is announced on an annual basis.
It is listed as a flag flying day by UK government guidance.
The Union Flag is specified.
https://www.gov.uk/guidance/designated-days-for-union-flag-flying
The official birthday of HM The King is also listed as a flag flying day by Scottish government guidance.
The Scottish Saltire is specified.
https://www.gov.scot/publications/flag-flying-on-government-buildings/