I have been able to connect HDMI monitor using high quality cable 20276 with the latest Intel® HD Graphics 3000 driver version 4101.
I would recommend contacting your computer manufacturer for possible warranty options on the HDMI ports.
Do you think it is a hardware problem? I don't think so, in Windows 8 works pretty well. The problem is when I used Windows 8.1...... I think the Intel driver has still some bugs on Windows 8.1.....
The driver versions 4101 has been working on my computer using HDMI connection. I recommend contacting your computer manufacturer for further assistance.