First step is I'd make sure the lamp fan isn't an AC powered fan, because most all computer fans are DC powered. If it's AC you're SOL (it'll burn the fan out) unless you have a separate power supply (wall wart) for the fan.
If you're sure it's DC though what I would do is first find out how much voltage the old fan was drawing, in a computer the power is regulated such that it delievers a voltage apt for a smaller fan, in your setup it might have much more voltage for that fan. If that's the case then it'll rev your new "quiet" fan up to such a level it'll probably also be loud.
All that being said, if all is good, I'd simply strip the 3 wires, use a 9v battery and hook up one wire to positive one to negative. If it spins you're in business those are the two wires, if not try another pair. Once you figured that all out simply make sure the fan is spinning the right direction (ie air blows out the the side with the sticker), if it doesn't switch the wires around
So as long as the first two things don't happen you should be good to go