The Win7 laptop I use every day has resolution well under 1920x1080 so this shouldn't be an issue.
Try the most basic things first:
- check that the board has power, one of the LEDs between the two Type A USB connectors should show a heartbeat.
- check that the USB cable is plugged in correctly (you need a common USB Type A to microUSB device cable).
- check that Win7 detects that there is something connected to a USB port. Open "Device Manager" and look to see if there is an Android device, or something under COM ports. You should see something, possibly with a yellow triangle.
- if you see the Android device, then your problems are not with the drivers.
- if you see yellow triangles, then the drivers are not correctly installed.
To be honest I remember that getting my Win7 laptop setup the very first time required me to do something strange, but once it is set up you never need to do it again, so I don't remember what it was I did to get it running.
Full Disclosure: I am an employee of Qualcomm Canada. Any opinions expressed in this or any other post may not reflect the opinions of my employer