That depends on the quest you are choosing. If you are playing an Adventure Quest in an area that’s an easy walk from you, then, no, you can simply walk there. If that is not the case, you will need to get to the play area somehow, whether it be by car or public transportation. Some quests may take you to different locations within a broader area that is simply too large for the average walker.
Most Discover Quests, on the other hand, are designed primarily as driving tours.