If it were possible then the datasheet for the DS3231 would say so. In as much as I trust you to have read the datasheet carefully I'm going to say no, it is not possible. What you need to do is implement a conversion in the software that reads the date from the DS3231. AFAIK, that is your only option.
Unless there is such a thing as an Islamic Semiconductor Company which could design such a chip, and then have it fabricated by TSMC or a fab in Penang. If there isn't such a company maybe you could start one.
I am curious about one thing. Does the Islamic calendar have adjustments like Leap Years and Leap Seconds to keep the calendar in sync with the movement of the planet?
Thanks papabravo you rely helped me.
I actually found a way to convert the date format,
1# first we need to return the amount of days passed from 1/1/1970 using the function now() (from the arduino IDE )
2# once we now the amount of days passed, and we also now that 1/1/1970 is = to 23/4/5730 in the different date format
than we can calculate from 23/4/5730 + the amount of days we have, and then we would get to the date we are up to.