Do I need a special bike for track cycling, or can I convert my road bike?

You need a proper track bike. Road bikes cannot be safely converted for velodrome use — they have the wrong geometry (especially bottom bracket height), brakes that are not permitted, and horizontal dropouts are needed for proper chain tension on a fixed gear. Most velodromes will not allow road bikes on the track. Many velodromes offer rental track bikes for beginners, which is the best way to start before making a purchase.