Actually, not at all.
To start with SE isn't a forum. And well there's no way to kinda have a 'private' Q&A as a subset of SE. There's a few attempts like "Facebook.stackoverflow.com" where the subdomain was a page that showed stuff on a specific tag.
Practically speaking there's also nothing in it for SE to allow you to use their infrastructure and userbase to build your site with, well, pretty much nothing in return.
If you wanted to run your own Q&A site - sonic's comment links clones that could work for you but
So for example, if I visit forum.example.com, the page would be generated / hosted by Stack Exchange and use Stack Exchange's network but on my website only.
Is not a thing in terms of being a forum, or having subdomains for users, or somehow filtering specific posts for a specific view on a specific subdomains.
SE does have an API - but I'm not sure if the SE model fits your needs very well. I don't see how it would work in terms of the licence, or how such a UI would work. There's an old defunct project for a "terminal web browser" friendly SE front end - which is the closest thing I'd think of to a different SE front end. It isn't a 'private' forum leveraging on our userbase though.
And using SE as a backend for your own projects isn't something that feels like something that's good for the community or the network either.