Presuming the motors are 12V, and the 12V power supply is there to power the motors, yes, you can just disconnect whatever leads there are to the motors already and jumper some wires from the PSU to the motor.
You need to be careful because:
1) the motor will run at maximum speed (and this may be faster than you've seen it ever operate)
2) The motor may rotate in either direction
3) There will be no limit switches to stop the motor from driving things too far, or continuing to drive stuff when it's jammed.
So, make sure that you only apply power for a very short time and that you have nothing where it can be caught up in the motor or anything the motor activates.
For 12V, I might consider having one lead that I manually brush against the PSU output. If I see big fat arcs, or the motor does something extremely odd I can probably detect that before something bad happens.