You'd have to probably have one of a few options:
1) a suitelet with no login required.
2) a link to a web form that creates a custom record and then a scheduled script to convert the Quote based on the information gathered in the webform.
Neither is particularly secure, though I would think risk levels would be relatively low.
The concern I'd have is how you validate that the person on the other end is actually the customer, and not an employee/someone who has something to gain from the quote converting.