Was the Prophet's migration (peace be upon him) to Medina in Muharram or Rabi'? And did the Jews use Arabic or Gregorian months in their religion?

The Prophet's migration (peace and blessings be upon him) took place in Rabi' al-Awwal. He departed Mecca on the 27th of Safar and entered Medina on the 12th of Rabi' al-Awwal, according to the more accurate opinion.

Regarding the beginning of the Hijri calendar with Muharram, the Companions, during the time of Umar ibn al-Khattab (may Allah be pleased with him), agreed to adopt the Hijra as the starting point for the calendar. They chose Muharram to be its first month because it was the first crescent moon that appeared after the Pledge of Aqabah, which was a prelude to the Hijra.

The Jewish calendar was originally based on lunar months, as evidenced by their fasting on the Day of Ashura. Some of them later converted it to a solar calendar.