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).