I found this motor from an old piece of equipment. When a button is pressed, the battery powered motor turns on. When I took apart the button, its functionality was to make two metal plates touch each other, creating a current flow. I'm wondering if I can hook up these wires to my Pi's GPIO pins and program a script that will fake the wires being connected. Is this possible?
A relay is your best choice, as it will connect the wires, and will require only one line. The script will have to power the relay to activate the motor. Connect the relay to the button's signal, and it will simulate the button when you close the relay.
No! You can use a relay.
I suggest you do some research before connecting anything to your Pi and learn a bit about circuitry.
There are hundreds of books and tutorials.