Of course I was accessing that site from my phone, so I couldn't see the specs.
If you're outputting 12V I would hope the current is fine, but what's the amperage rating on the AC adapter?
Voltage might be your issue here. Most standard cooling fans run off either 5 or 6 volts depending on their location in the computer. See if you can get actual specs on the fan from the manufacturer and try to match them.
I've never wired one of those myself but I do know a few things.
1. Does your AC adapter have the correct voltage and current to run the fan?
2. Is the fan a variable speed that requires being hooked up to a controller (ie computer) in order to work?
Answer those two first and it'll help you...