If I had four of these motors and attached them to the Arduino Nano on four separate digital pins or analog pins without transistors and a 9v power supply attached to the regular power supply input would that work?
Yes, but don't expect to win any races.
If you use
analogWrite(HIGH) on your digital pins (3, 5, 9, 10, and 11) your motors should receive 40 mA max at 5V (see http://arduino.cc/en/Main/arduinoBoardNano). So the motors will run, just not very fast.
A switching transistor would be a very simple addition that would allow you to achieve higher current input to the motors and get their full potential.
If you want to reverse the direction of the motors you'll need to investigate getting a cheap motor driver board.