Here is my observation after spending some time on this issue.
Mac Mini, Mid 2011
[OK] HDMI -> Asus PQ236Q DVI
[FAIL] Thunderbolt Mini-DisplayPort -> Asus PQ236Q DisplayPort
MacBook Pro 13-inch, Mid 2009
[OK] Mini DisplayPort -> Asus PQ236Q DisplayPort
Based on my setup, Thunderbolt Mini-DisplayPort connection doesn't seem to be fully compatible with ASUS PA236Q DisplayPort input. The best workaround would be to use HDMI to DVI cable.