If you want to fit into the Moodle quiz/question engine, then you need to follow how they are designed to work, not how you think they should work.
So, if you want that sort of interaction, just make you quesion type work well with the 'Interactive with multiple tries' and the 'Adaptive' options for 'How questions behave'.
I now remember one question type that does use Ajax: the STACK question type. However, it only uses ajax for validation. The Check button still reload the page, and the question type works fine if you disable JavaScript.
It would be possible to change the question engine so that, with Adaptive or Interactive behaviour, all questions get updated in an Ajax way, but that would need to be a change to the whole system, not just one question type.