I had no problems accessing this site before I installed Avast, so it can't be a browser issue.
This really has nothing to do with avast:
1. avast! doesn't monitor https traffic.
2. avast doesn't check site security certificates only harmful content.
3. This is a browser issue as the warning comes from your browser. You need to look at the security settings relating to site security certificates and either lower the options or see 4. below.
4. It is quite possible that the site has had its security certificate revoked since you last visited it. Or the other computer has different security settings, but I would have thought that there would have been an option to ignore the warning and connect to the site, assuming you knew it and were confident it was OK.
If this was a warning from avast you would know about it, as the first time you have an avast virus warning it will scare you to death, very loud audible warning, but it doesn't warn about site certificates.