Will the Garmin operate on battery when it is unplugged? If the Garmin battery is completely dead, it may not work even when plugged in until it has charged up to a minimul level.
A couple of other things to check ... Does your garmin power plug have one or more LED lights? If so, the lights should be on if it is receiving power. There is a little fuse inside the tip of the Garmin power plug. You could check to make sure this fuse isn't blown. Also, try both the accessory outlet and cig. lighter outlet on your Elantra to see if that makes a difference. Other than that, the only thing I can think of is to make sure the power plug is completely seated in the cig. lighter outlet.
By the way, what model Garmin do you have? I have a nuvi 755T and 855 and participate in several Garmin forums.