LOLOL
The facebook profile is probably about 10 times more valid of a method of checking someone's background than a polygraph. Actually, nearly infinitely times more valid - as there's very little validity to polygraphs. (By the way, has a polygraph test ever caught a spy subjected to routine polygraph tests? Answer: no.)
Depending on what part of the county government she's working for, there may be a LOT of reasons for such a background check. i.e. suppose someone were working for the sheriff's department and it wasn't until the prosecutor lost a major case for technical errors that someone realized the person in the background making the blunders was a member of a gang. Now, I'm sure your wife isn't a gang member, but there are plenty of other reasons.
The nice thing about it is - at least the county government is being honest about it. There are plenty of ways of getting that information without your knowledge. Regardless of your "privacy" setting, if you post it online, it's out there.