Does anyone know how ASQ exam requirements are verified?

I believe I meet the requirements to sit for several ASQ exams, but I'm not sure how much of my work experience is verifiable with regards to areas of BoK. Surely they aren't just going to take my word for it.

ASQ might accept your work history. I seem to recall that I had submitted something from my employer's HR department when I applied for my first exam (back in the Stone Age), but I doubt they would do a check (or ask for more info) unless there was some other potential red flag... I was always under the impression that ASQ didn't want people applying that were likely to simply fail the exams due to professional immaturity.

If there is an additional (mostly harmless) gatekeeping reason behind the requirement, it could be that because ASQ does salary/career related surveys, they want to have some idea of who is opting to join based on their backgrounds.

I don't remember having to submit any documentation on my work history. However, I was busy dodging dinosaurs and may have forgotten.

A letter on company letterhead from the HR department that states your employment status (full-time, part-time, etc.) and a general list of your responsibilities/duties is usually sufficient. Of course, it should be signed by someone in the HR department.