The phrasing is a bit awkward, to be sure, but I don't think you've got the gist of it with your proposed replacement.
Imagine an office with 3 people: Elvis, Ed, and Mike. Mike manages Elvis and Ed. From your sample dialog, Jennifer assumes that the survey asked Mike what Ed and Elvis do that Mike found annoying. Rob clarifies that the survey actually asked Mike what he thinks Ed does that annoys Elvis and what Elvis does that annoys Ed.
As to the grammatical structure, the grammar of the given phrase is correct (if, as I said, a bit awkward), while your replacement has a few issues:
"managers believes" -> singular/plural mismatch; must be "managers believe"
"believe in" -> is used to indicate a conviction that the stated object or idea is real or true; to "believe in things that make people annoyed" is to hold the position that some unspecified things exist that make people annoyed, which is a position so obviously true as to be not even worth mentioning.
"what managers believe in things" -> this is a statement selecting a subset of managers, not a statement about annoying habits; it is equivalent to (which managers are there that believe this), not (what thing do these managers believe).
"make their workers annoyed each other" -> each other is left hanging here; you need a "with" or an "at" to make it grammatical.
A better replacement phrase might be "it was about what managers believe their employees do, that their other employees find annoying."