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The Hajj is during the known months, so whoever executes the Hajj in them
shall refrain from sexual contact, wickedness and arguing during the Hajj.
2:197
As the above words clearly indicate, God allowed Hajj to be observed any time within the known months. These are the four months which God called "Hurum" (9:36). However, the Muslims today have ignored God's merciful concession and have confined the Hajj to only the first few days of the month of Zhu Al-Hijjah!
For full details of how to observe Hajj according to the Quran, please go to the following page: Hajj, the Quranic way