Not easily, no. This is a separate record that has a many to one relationship (you can add multiple user notes to a single record) so already you have the challenge of looking for at least one user note versus just making a field required. Maaaaybe you could do some sort of scripting, but even there, I think the user note record doesn’t actually get created until after the parent record (and then subsequently attached/associated with it). If I’m correct in this assumption, that means that you couldn’t actually prevent saving of the parent record based on the user notes, since it won’t get created until after the parent record is already saved.
My recommendation would be to just to make an existing field mandatory (e.g. memo) or create a custom text/long text/text area field and make that mandatory. Or if for some reason your specific use case really needs user notes I’d do a detective control via saved search. All of those should not be too hard, and waaaay easier than trying to figure out how to make user notes mandatory via script.