No you do not need live in a big city to do well as a graphic designer. The only advantage is having access to lots of clients you can court in person - well planned visits you can be scheduled even if living in a small town. Today’s virtual world allows for people to work anywhere as long as connectivity is good. Publicity and public awareness is key post work on social media (Facebook & Instagram) and on design sites like Dribble and Behance along with a well crafted site of your own. Good luck!