Is it possible to track Conversions of unknown Tagged URLs?

When a user is navigated to your site with utm parameters within URL so since google analytics code present on your page, it reads the parameters and record the data in google analytics.

Since you don’t own a url that means your tracking code is not present within url code and hence no data collection within GA.

If these links are present on your own site, you can track them as outbound links within event when links are clicked.