There is no “API” for quick forms, no. But you may be able to populate and submit the form programatically if you can get the CSRF token that Drupal’s Form API generates. I’ve never tried it though.
Broadly speaking, quick forms just create records that follow a convention, and you can create the same kinds of records yourself via the API directly, without going through the form.
This would be part of the value of documenting conventions that I’ve talked about in the past: Documenting conventions