It's hard to see the 4-pin header from the angle of your image but I also own the Gigabyte Gaming 7 mobo. I understand what you're trying to do but from looking at the manual you're only going to get 5V from the header connection, enough for I/O shield lighting but probably not for your entire case.
I have the Phanteks Enthoo Luxe which has exterior lighting built in to the case. It requires a SATA power connection to the PSU in order to light up both sides of the exterior.
What case are you using? Case manufacturers often release their own LED strips which ensures compatibility however there are 3rd party options out there (which seem to use SATA or their own separate PSU). For illustrative purposes, this (Australian) website has a few options collected together which you can look in to:
https://www.pccasegear.com/category/1354_1356/modding/led-lighting
some 3rd party options:
https://cablemod.com/products/?filter_product-category=86
http://www.phobya.com/eng/phobya.html