Write a custom AEM Sling Servlet and use JSON data object to build JSON data. See this AEM Community article:
https://helpx.adobe.com/experience-manager/using/custom-sling-servlets.html
Once you have installed the OSGi bundle - other clients can invoke it using HTTP requests.
Here is a brand new article that shows you how you can read the JCR and then convert JCR data into JSON. THis is a technique that you should know about as an AEM developer.
https://helpx.adobe.com/experience-manager/using/quiz.html
Simply using a JSON extension on a AEM URL is not going to give you the precise JSON data that you most likely want. Its better to write app logic in an OSGi bundle that will create JSON you need.