How to count the number of calls to a rest service?
Do your REST service needs authentication?

If yes, you just log each access attempt by users:
One attempt per one record per user or one record per user and updating the count column always.

If there is no authentication, you will have harder time to achieve this. 
It would be also good to know HOW they are making this rest calls (eg: clicking on a button which resolves into a rest call?).

You could still try to check session and find out which user made the requeset.