The inductors in a graphics card are for the most part used in its VRM, or voltage regulation module. VRMs are a type of DC-DC switch-mode power supply, usually a synchronous multi-phase buck regulator. This means that it uses a bunch of "phases" in parallel, each consisting of a pair of transistors, an inductor, and a capacitor, to change the +12V input from your PSU to 1.xxV for your GPU to use.
In order to supply a different voltage or different current the transistors switch on and off at different frequencies. If the GPU demands more current, then that will cause a voltage drop across the inductor, which makes the transistors switch faster to keep the voltage where it needs to be. And the reverse. All this happens in microseconds.
An inductor stores electricity in a magnetic field. This magnetic field pushes and pulls on the ferrite core used in some inductors. When the voltage across the inductor changes, the magnetic field changes, and the inductor moves slightly. When the voltage changes very, very quickly (as is the case when switching transistors are changing its voltage thousands of times a second) the inductor vibrates. And that vibration, if it is between 20Hz and 20,000Hz, is audible to the human ear. Hence whine.
The frequency at which the switching transistors operates varies, but it's generally between 10,000Hz and 100,000Hz. If your GPU is whining, the transistors are switching at between 10,000 and 20,000Hz, or a primary harmonic of those frequencies. So if you want to stop the whining, you need to make the transistors operate at a different frequency.
So you change the amount of power drawn. Overclock and overvolt your GPU and the transistors will need to switch faster to provide power to the GPU, and if you're luck you'll bump them from 17,000Hz to 24,000Hz. Underclocking may work as well. And suddenly your GPU will whine no more.