The Intel board DQ67EP has been designed to work as a desktop system with a real monitor attached and under a desktop environment like Windows OS.
You could give it a try setting your operating system to never turn the monitor off (automatically), hopefully it will fix your problem.
You can check the operating system compatibility list on the following link:
I own a DQ67SW motherboard with BIOS SWQ6710H.86A.0055.2011.0921.1044. It is a headless server runing debian squeeze 6.0.3 and I have pretty much the same problem.
I found a workaround, blacklisting the i915 module seems to solve the problem, for the moment.
Use kernel parameter drm.debug=0x06 for more info.
My bet is that i915 driver detects nothing plugged on any connector (vga/dvi/etc) and does some kind of power management which messes up things, but this behavior only shows on module loading. Plugging a monitor trigger some event on i915 module and kvm becomes available again, you can unplug monitor and continue using kvm.
(sorry my poor english)