All I can say after research that there is no such thing there as areas for camping - you can set your tent anythere in safe, and relax.
But be aware to the rules of National park. There are general things there, but I suggest you to check this on arrival. During summer in Kopaonik some hiking tours are available from various firms, and, maybe, you could find any information about camping there.
According to a few Serbian websites, camping (setting up a tent) is not allowed in Kopaonik NP. A few Serbian campers mentioned setting up a tent a bit far off the road in the woods (being aware it's disallowed).
Further search found this inn/restaurant which includes camping grounds. It is located in Vlajkovac, 244th km of the road from Belgrade (that would be the road to the north -- red line on the map).
Note: no personal experience with camping in Kopaonik, so don't take this as a recommendation.