Are the months of Hajj mentioned in the Quran Shawwal, Dhu al-Qa'dah, and Dhu al-Hijjah, and is it permissible to perform Hajj during them?

The months of Hajj are Shawwal, Dhu al-Qi'dah, and ten nights of Dhu al-Hijjah according to the Shafi'is. For the Malikis, they are Shawwal, Dhu al-Qi'dah, and the entirety of Dhu al-Hijjah. The Hanafis and Hanbalis hold them to be Shawwal, Dhu al-Qi'dah, and the first ten days of Dhu al-Hijjah. All concur that their beginning is at the start of Shawwal, and one cannot commence the rituals of Hajj after the dawn of Eid day due to missing the standing at Arafat. The intent of Allah's statement, "Hajj is [during] well-known months," is that these months are the time for entering into the state of ihram for Hajj, and it does not mean that all Hajj rituals can be performed at any point within these months, for each ritual has a specific time before which it cannot be performed.