It's been less than two weeks since I last had to re-login and wait for the verification code. A process which I thoroughly hate (putting it mildly). I checked the box to ensure that I only get asked this every 30 days. But I was asked to do this again today after having done it last on the 9th of December. Today being the 23rd of December. My bank does not make me do so much to use my online account. You should have a special something which can identify people who have never tried to break into your (non-banking) system. And who can be trusted. QRZ is not a bank hence I don't see the need for all this verification code business.
The web site is (presumably) using a cookie as a verification token and it has an expiration date. If you are going to do it that way then an expiration date is (from my dated experience) actually a pretty good idea. Yes, it is an annoyance but at least they are trying have a bit of security. On the list of things to fix I would focus first on the annoying habit of redirecting you to the main page after login rather than back to where you were. But back to the original issue I would suspect that an improvement in the login would involve a substantial change in the underlying system and/or purchasing of expensive tools.
Please file a direct support ticket: https://ssl.qrz.com/support HQ support folks may be able to sort out what's happening. Dave W7UUU