NO, you cannot use the PICAXE output directly to turn on motors, relays etc.
A PICAXE output can supply (source) or sink a maximum of around 20mA.
That is okay to drive a LED with a series resistor but not much more.
Depending upon the exact fan (size, air flow etc) it could draw anywhere from 100mA up to 500mA
You will need that MOSFET transistor as a form of solid state switch to control the fan motor.
If it is a straight on off function then even a bi-junction transistors driven into saturation, so that it does not dissipate a lot of heat, could do the task.
Are you sure that that PC fan is 5Vdc? Some are, but many PC case fans are also 12Vdc.
Make sure you use a separate power supply for the fan motor with the 0V connection linked to the PICAXE 0V as a signal reference line. If you try to run the PICAXE and the fan motor from the same 5V supply when the motor starts there can be a significan drop in voltage which might reset the PICAXE so the program appears not to be running properly.
