I found the answer to your question....q.com accounts can be established as a POP account or as an IMAP account and when you enter in your information for the account set up in settings it automatically defaults to a POP account setting. But q.com accounts need to be established initially as an IMAP account in order to be able to see existing folders established within the account. The current IOS does not allow q.com accounts the option after initial set up for an IMAP configuration, so you need to do the following steps to bypass the system and configure the account correctly
In settings go to Mail, Contacts, Calendars
Select Add Account
On the next screen select Other for the account
Select Add Mail Account as the email account type
Enter in your email address
Enter in an incorrect password
Attempt to establish the account and after a few attempts the system will default to a page in which at the top of the screen you can choose IMAP or POP configuration
Select the IMAP configuration and enter in the correct password and follow the steps for the rest of the account set up
You should be good from there, sounds a bit extraneous in order to establish an email account but it is a successful work around. Hope this helps and good luck. :)