Yes, that sounds like it could be coil whine (coil whine doesn't have one specific sound, but I'd say this is one of them).
As for fixes, lower FPS sometimes lessens the effect, so for instance if your monitor is 60hz use a framerate cap. AMD has FRTC in their drivers, I can't recall what nVidia has - I seem to recall using it on a GTX 460 though so I think nVidia does have it.
Some PSUs can cause it when in combination with some GPUs, ones with lower ripple tend to lessen the likelihood of the effect I think.
Some people say that if you find a program that has 1000s of FPS (such as the menu of a game, etc) and leave that running overnight it can cause it to stop once the card burns in a bit. Sometimes just normal use will cause it to stop.
I've had 2 GPUs in the past where it was just a fact of life though, nothing fixed it. Both luckily weren't too bad when the side of the case was on - yours on the other hand seems like it is pretty loud given the distance you had the camera at in one section of the video. If it is indeed extremely loud I've heard of getting a replacement card, I certainly wouldn't be happy if it was noticeable on a high value card.