This is sort of true. If there is something wrong with the battery then there is a good chance they will fix it for free even if it's not under warranty, rather than billing you several thousand dollars.
But if, say, the dash lights come on saying you need to take it in immediately because there is a problem with the battery, but when you take it in they can't find anything wrong with it, then they will still bill you a few hundred bucks for looking at it.
The system should totally keep a record of the dash lights shown, and they totally shouldn't charge you if you told the truth, yet they couldn't find anything wrong.
But if, say, the dash lights come on saying you need to take it in immediately because there is a problem with the battery, but when you take it in they can't find anything wrong with it, then they will still bill you a few hundred bucks for looking at it.