This is probably something you'll need to implement yourself as it is too-specific a requirement to have been included in the JDK's API.
Assuming you've developed a standalone Swing application you could consider using the Preferences class to store the number of times a user has printed a document for a given date. On Windows this translates to storing information in the registry and is therefore "hidden" from the user to a certain extent, but would allow you to reset the value in an emergency using regedit.
The advantage of this approach is that the user cannot circumvent the print-threshold by simply restarting the application.