To change the primary email address associated with your Quora account, go to https://www.quora.com/settings.
Click on 'Add another email address' and input your new address. After clicking 'Add Email', you'll receive a confirmation email to the address. Click on the confirmation link you'll receive, and go back to https://www.quora.com/settings to set it as your primary email by clicking 'Set As Primary' next to the address.
You can then delete the older address from your account if you wish.