Hi @bsr060893,
Could you please confirm if your custom image component has a Sling Model and Exporter is enabled on the same via annotation.
Below link explains how to enable a JSON export for a component.
https://docs.adobe.com/content/help/en/experience-manager-65/developing/components/json-exporter-components.html
There is nothing like default exporter.
XF page or Site page is accessible via "model.json" is because the respective page component has Sling Model with Exporter. Also the container components used within in turn has its own Sling Model with Exporter.
Given your existing set up, the only possible way to expose XF or Site Page as JSON would be to make use of JSON rendering of Sling Default Get Servlet. - "tidy.infinity.json or tidy.-1.json" which will expose the entire node structure with utmost depth. (To avoid custom implementation)