The answer is maybe.
Usually the 4*AA holders are used to create batteries by serially connecting the 1.5 V (nominal) cells into a 6 V battery. The voltage drops with use and can go as low as 3.2 V in some cases when the battery is nearly depleted.
On the other hand the fan in its specifications has the 12 V input voltage and maximum current of 100 mA. It's more or less usual laptop fan and it may or may not start working at 6 V. Basically such fans need some voltage to start operating and then can work on lower voltage. The 6 V provided by the holder populated with fresh AA cells is on the lower border of the start voltage. If the cells aren't fresh, the fan probably won't start.
The solution to this problem is to get two such AA cell holders (or one for 8 AA cells). This would give you nominal voltage of 12 V at the fan and it should work fine.
About the connections part: Take two holders, connect red wire of one holder to black wire of the second holder. This will give you a 12 V battery and you'll have one red and one black wire free. Connect the black wire of the newly created battery to the black wire of the fan and the red wire of the newly created battery to the red wire of the fan. You may want to put a switch on the connection between the two red wires or on the connection between the two black wires so you can more easily turn the fan on and off. In addition to that, the lowest voltage you can expect from the battery now is around 6.4 V with no load. This should be enough to be sure that the fan will work until the cells are completely depleted.
And yes, you can safely cut the white connector from the fan, it's just there so you can plug the fan into a device which has the matching receptacle. If you don't have the receptacle, you don't need the connector.