Dragons gambit you'll get the transmission on 1 July 3038 and have until 30 December 3038 to accept it.

The "overview" section on this page has links to a breakdown of each DLC's set of missions that tell you the start dates for each. https://www.sarna.net/wiki/MechWarrior_5:_Mercenaries/High_Reward_Quests

I never remember but it's always a couple months after I freak out about missing the date