Is there a reason this doesn't exist?
Because it's really fuzzy, and HNQ status is not tracked or logged. Check out Please open the Hot Network Questions to auditing via the Data Explorer for an overview of the issues here, and also follow the links in that OP and beyond for some more info about the algorithm.
Should we be notified if we make it onto HNQ?
I don't think so, because there's nothing you can really do with this information. It's merely mildly interesting, but it isn't actionable. See, you write:
I was totally oblivious to the fact that this had happened and thought it would be cool if SO would notify us ...
And I think it would be cool, too, but ask yourself: What would you have done differently had you not been oblivious to that fact? There is really no action to be taken here.
This comes up again in your comment:
... I wonder how many users have wanted to make the list and not been aware when they have.
Again, it would be neat to know, sure, but what would those users have done differently if they were aware beyond pumped their fists?
In any case, it's relatively easy to tell if your question ends up on the HNQ list, you'll generally notice an unusual increase of vote/comment/answer activity, which do show up in your inbox. You'll also see the view counts very high relative to other questions, as you noticed (and at certain points the badges will show up in your inbox as well). You can also look for your question on the sidebar or in the hot tab at https://stackexchange.com/questions?tab=hot.
... I wonder how many users have wanted to make the list ...
Note that getting a question on the HNQ list isn't necessarily something to strive for, it's just something that happens. What you should do is just always try to ask good questions. If you end up getting one in the HNQ list, that's just a bonus.