And the screens are LED? The two major "noise from pickups" comes from the CRT screens and fans. But, I have never in my life had an issue with a computer causing pickup noise--how near are you to said computer? Does it go away when you walk to the other side of the room? If not, it's either not the computer or not pickup interference--but, potentially a ground loop in the system.
Move around the room.
Turn off the monitors. (VGA)
Try a different guitar cable.
If you have a DI that will allow you to lift the ground, employ that.
Use a humbucker pickup.
Did any of these make it better or worse?