My time machine backup won't show up?

My MacBook Pro has been really slow so we went to the genius bar and they said we needed a new hard drive because it's about to give out. We are now trying to back it up to get the new hard drive. My mom has an old Time Machine backup device that she used to backup her computer to her MacBook Air. We plugged it in and have waited for an hour and the MacBook won't recognize it. Nothing is showing up in the finder or desktop.