I understand cultural significance of Hijri but its use on the web will be misleading to many. However you clearly have a particular audience in mind so please read on.
If you find a solution, then ensure the calendar is annotated to indicate that is is Hijri - unless that is absolutely clear from its context. Even consider including a Hijri/Gregorian switch so users can choose which to use (tricky to implement but I'm sure someone has done it).
I've not though about this before but I guess the Unix Epoch itself (1 Jan 1970 Gregorian, or 10 : Shawwal 22 1389 Hijri) is actually calendar-neutral (or maybe competes with both calendars), because it is based on yet another "temporal zero" ("temporal false zero" at the risk of being universally provocative).
If you're in one of those predicaments where cls.__private attributes just aren't enough since they can easily be accessed through inst._cls__private, and you need something a little more secure, here's ...