Hi there,
My project is connecting a 12v digital counter to a 110v sewing machine motor. I've attached the photos of the counter.

Your pictures are tiny thumbnails and are no help. If I understand correctly you would have to tap into the 110 AC of the motor and convert it to 12V DC to the counter. If you have a spare power brick (wall wart) from an old unit that doesn't work, it may supply your 12 volts for you, but you have to be sure of the DC polarity (positive is usually red wire and negative black on the counter). The wall wart will not be as easy to determine which is which unless you have a volt-ohm meter.

What are you trying to count? Revolutions of the motor? Stitches (up/down travel of the sewing needle)? In any event, the counter will expect some sort of sensor input which provides a short circuit/open circuit or 4.5-30V pulse with every cycle of whatever it is you're counting.

You can find a whole variety of sensors on the Omron web site:

http://www.ia.omron.com/