It doesn't look like it'll be a problem. Those SATA ports appear to be slightly above where the GPU would sit. Worse comes to worse you could rearrange where the cables currently are and use the open ports.
I'm not even sure what that red cable does let alone the black one next to it. I tried to follow it to see where it plugs into, but the manufacturer bound and tucked all the cables away so I can't follow it.
Would I need to change the BIOS if I plugged the red wire into a different SATA port?
It'd go to either a HDD or an optical disk drive. You shouldn't need to change anything in the BIOS. Another concern would be your power supply. GTX680's require 2 6-pin PCI-e power cables at the least if I remember correctly.
Ry, the red cable is a sata cable it provides the reading abilities to the device whether its a CD/DVD/BR drive or a HDD, SSD. And without knowing your power supply it will be hard to say if you can handle a 680 what power supply are you currently using Ry.
Ry, you would not need to go into the bios if you plugged something into a different sata port.