I think the answer is No.
If we try and compress these atoms, we end up (eventually) with the nuclei close enough to be forced to fuse. Fusion would mean we've formed a single nucleus.
This stage is unavoidable.
So your two atoms question now reduces to whether a single nucleus can form a black hole ?.
A nucleus is a kind of complex quark-gluon mix and if we compress it more we end up with a very dense version of that which we basically don't have physics to model properly.
It's extremely unlikely that conventional general relativity can be applied to something that will be so small it's actually smaller than we think we can apply quantum theory. And the energy density involved at that point would be so high our current theories don't make sense any more. We need a quantum theory of gravity to do this and we don't have one that works well enough. In fact we're not even sure a quantum theory of gravity would allow us to go to such small, high energy scales - even that is unknown.
So we're in uncharted waters.
So why "no" ?
Well, to force such a compression of a nucleus we'd have to apply energies to a very small region of space - smaller than we think it's possible to do, because of the consequences of the uncertainty principle. Put simplistically, beyond some point we'd not be able to simultaneously say where the nucleus is and how fast it's moving. It would be impossible to confine to a smaller region. This would happen long before we reach the Schwarzschild radius, at around the Planck length.
As you'll see from the answer by @James-K , the Schwarzschild radius is about 10−53 m, but the Planck length is 18 orders of magnitude larger at about 10−35 m.
So we could not realistically confine and compress our nucleus into a small enough space to ever reach its black hole size.
Now we can make a generic catch-all statement that a new theory might provide some loophole that lets us get around that, but it does seem unlikely as we'd expect a new theory to reproduce most of what we already know at those limits. It's hard to imagine the uncertainty principle "going away" so I don't see a way around that.
There's an unproven possibility of a yes.
A quantum theory of gravity that works might (repeat might or might not) find that gravity at that scale changes its character and allows it to form event horizons at larger sizes than we'd currently expect for such mass-energy ranges.
But we lack any evidence to support that idea, and I won't convert a "no" to a "maybe yes" simply to allow room for any wild idea. That's science fiction, not science.