Well that is an older card and it's unlikely that it has 0dB fan feature. This is a relatively new feature.
However that link is directly to nVidia (who don't actually sell cards directly), not the manufacturer. There are probably at least 10 if not more board partners with nVidia for that series of GPU's. Some cards will be completely reference cards (nVidia spec, PCB, firmware, etc) and some will be customized. It's these custom features that things like the 0dB fan falls under. This feature isn't in the nVidia reference firmware. So I'd need the link to the actual card from the manufacturers site to determine what features the card has.
Again though, it's unlikely that this feature was around in a card that old. Like I said, it's a fairly new thing that ASUS, and MSI (there may be others I don't know of) have come up with to distinguish their cards from their competition.
So if I'm right in saying this card doesn't have that feature, then there is only one conclusion that can be made about your fan not spinning. Assuming its not a driver or software issue, then the fan is failing. If it's the fan, then it probably stops spinning when the card is cooler and the fan controller is trying to spin is slowly. Due to friction from a failing fan bearing, the fan can't overcome the friction to start spinning. Once the card gets warm enough, the voltage to the fan gets to the point that it's able to overcome the friction in the bearing and start spinning. If I'm correct, then this will only continue to get worse until the fan no longer spins at all regardless of the control speed.
There is a simple test to test the bearing in the fan. However as easy as it is to do, it's difficult to explain. Basically with the computer off, you gently spin the fan with you finger. It should spin for a little while before stopping if the bearing is good. Also with a good fan bearing, you'll notice that when it does stop it will sort of rock back and forth between two positions before slowly coming to a rest. This is due to the magnets in the fan. However if the fan bearing is failing, then it will stop spinning fairly quickly and it won't rock back and forth at the end, it will just stop.