Without knowing the model, it's difficult to answer. Most fans are held in place by simple screws, but other models have assemblies with special screw-cum-springs or pronged plastic holders that either require special care to be removed, or can't be removed (once locked, they need to be broken and replaced).
If you have normal screws with or without washers, you just need to be sure not to lose them as I usually do :-). The fan is very likely to be a brushless fan: you can probably even replace it altogether with a model of identical voltage and equal (or less) current rating.
Otherwise, the common models have a hub cap that's flush with the surface and can be removed with a small blade or sometimes a fingernail (it can be covered by an adhesive reporting the fan's data). Blow air into the uncovered recess, and place one or two drops of light machine oil inside (WD-40 is also good). If you have an air bottle with flexible needle tip, you can blow "inside" the fan, under the rotor, to remove some dust. Replace the cap and, while not as good as new, the fan should be a little quieter.
Also be careful to blow air into the heatsink to dislodge the dust that usually accumulates under the fan. The heatsink is probably good; if not, replacing it would almost certainly require also reapplying thermal grease. If you're familiar and comfortable with the operation, go for it (also because some cards are equipped not with proper thermal grease but with thermal pads; in which case, replacing them with a slightly greased copper shim of appropriate gauge will improve things considerably). Otherwise, my advice is not to risk it - improperly applied grease can destroy the card through overheating, and you can find some servicing shop where they'll do that for you for much less than the cost of a replacement card.
If you really care about silence, there are special assemblies (but check for available space) that use an oversized heat exchanger, driven to high temperatures through the use of a Peltier cell. This way, if properly installed, the heat exchanger is capable of getting rid of the heat through passive convection, with no fan at all. Not all desktops can use this solution, though. Other similar solution do not go the whole way, but are able to employ a smaller fan.