That PIR looks very similar to the one in the tutorial I linked to, so should be OK.
If you already have an Arduino, you could build up the PIR side and the motor side of your project using the Arduino (based on those tutorials) to prove that things are working ok.
You could then try removing the Arduino. Your PIR should output a "high" level on its output pin when motion is detected. If you dropped the Arduino and connected the PIR output directly to the resistor in the motor circuit, the "high" will turn the transistor on and start the motor. I think that is the logic you wanted? Motion = motor on, no motion = motor off. You will still need to supply power from the Arduino (as per the tutorials) or substitute another 5V supply, and keep the arrangement of motor, transistor, diode and resistor the same.
One point to note, though. The motor tutorial uses an analog PWM output from the Arduino to give variable speed motor control. The output from the PIR will only be a steady high or low. When it is high, the transistor will be switched on (not alternating on and off quickly as with PWM). So the motor will only run at one speed when on, which may or may not be what you need.
All the best
Ray