As previously mentioned, all you need to do to participate in a PQ is to be in the area. Just being there, however, does not mean that you will be able to accomplish the goals to advance the quest. In most cases, the first stage of a PQ consists or killing mass amounts of a certain type of mob that spawns rapidly and roams the area. Stage one normally isn't on a timer, so you can take as long as you need to complete this task, which, with time, can be accomplished by any solo player.
Stage two, however, is generally much more difficult. Usually stage two will consist of interacting with items that pop up around the area. These items are usually guarded by one of more Champion-class mobs making this an exceedingly difficult task for just one person. A minimum of three people working together may be able to complete stage two, but a group of six or more is recommended. It is important, though, that the group consist of all three major play-types: Tanks, healers, and DPS.
Once stage two is completed, the third and most difficult stage begins. Usually stage three consists of killing one Hero-class monster. I know of no class that can solo a Hero monster, and even a tank will typically require two healers dedicated to keeping him alive. A minimum of a group of six players, and often more, is required to complete the final stage of a PQ.