If you're having no problems using VGA, then I wouldn't think so. Regardless, you can download a free (safe) program called CPUID HWmonitor that will show you all the temperatures that are monitored by your system.
In the case (pun intended) that your system is dusty and old inside, maybe it hasn't been opened in years, you really should think about cleaning it out with some compressed air, or even sending it in to a local computer repair store for maintenance.
You can scroll down to your graphics card and check the temperatures, then compare them with averages for your card (using google).
If you bought the computer from a big company (Dell for example), you may want to check the status of your warranty before preforming the next suggestion.
Another thing you could do to make sure your graphics card is running safely would be to pop open your PC and check the fans and chassis (if any) for dust accumulation. As you probably know, a build-up of dust can cause core components of a computer to overheat and function improperly. A graphics card that is functioning improperly may show anomalies on the screen, such as pink lines or distortions.
If you were having PC crashes from this problem, windows often will tell you that there was a problem with your graphics card when you boot back up. If it didn't say that, then the problem is more likely to be a connection based issue.
//other notes
I looked up your card online, it's pretty old. I've seen cards like yours that have been exposed to high temperatures for long periods of time, the boards will literally warp on them. I've also seen cards stop functioning completely from heat damage too.
//summary
- try to reproduce the problem with a DVI to HDMI conversion
- check temperatures with software
- check the inside of the computer (or have someone with experience), look for signs of dust and heat damage.
- consider replacing or upgrading your graphics card