There actually is a pilgrimage festival by Muslims to Jerusalem.
It is Nabi Musa, a seven-day festival celebrating Moses.
What is the date of the festival?
It turns out that the holiday is not pegged to the Muslim calendar, but to the Christian calendar. And it does not date back to antiquity, but to the 19th century.
Read this description from 1955:
Wikipedia notes, the date for this "Muslim" holiday is "beginning on the Friday before Good Friday in the old Orthodox Greek calendar."
In other words, the holiday is fake. It was meant specifically to counter Christian pilgrims converging on Jerusalem during Holy Week.
It isn't a celebration - it is a veiled attack on another religion.