I've checked my Rezound for the default browser, Chrome & Dolphin. None of them check or can be set (to my knowledge) to check for an expired security certificate on a website that could be affected by the Heartbleed bug.
Only Firefox (so far) seems to check for an expired or invalid security certificate on websites.
Prove it to yourself:
This web site was intentionally designed to have an invalid security certificate and should refuse to load (you'll get an error message).
If it successfully loads, you'll see a message telling you that your browser isn't checking for invalid security certificates.
I confirmed this behavior on my Hisense Sero 7 Pro also.