If they have no idea what you are talking about, you could ask whether the dealership will send you a letter saying you do not currently owe them anything, nor have they charged off anything you owed. This separates trying to figure out exactly how the letter arose from confirming that its claim is not valid. But your description raises many questions.
Are you confident the letter actually came from the dealership? Does the letter ask you to send a check to the correct dealership address, payable to the dealership, with all other contact info matching the dealership?
If not, e.g., if it asks for payment to a "loan servicer" that the dealership confirms is not theirs, or if it asks you to call a non-dealership number "for more information" (phishing?), that party is likely the one scamming you.
If everything does match, then unless it came from an unrelated party who enjoys wasting their own time, the main issue is whether the dealership is scamming you or made a mistake.
Did the people you spoke with indicate that they are aware of the type of letter you received and that they don't have a record of you being sent one -- or that they simply aren't involved and don't know about their employer's collection practices?
Did you offer to email or fax them a copy of the letter? (Maybe they would immediately say, "It's fake -- that's not even our letterhead style.") Did they validate that the name in the letter was at least an appropriate person at the dealership who would be expected to know about collection letters?