At the end of overtime, the official scorer notices that the home team scored one additional point in the 4th quarter before the officials leave the court and verify the final score (Visitor 62, Home 61).

The referee rules that once overtime begins, there can be no corrections to the previous quarter. Is this correct?

I agree that the home team can't be declared the winner in regulation once overtime starts, but if the error is spotted before the final score is recognized, can't the official add the point and play a second overtime if he feels the error is justifed?

Thanks.

Once the ball is tossed that is the end of any correctible errors.

What happens if it's just a difference between the book and the scoreboard?

NFHS Casebook 2.11.10 B
 

Correct the score and continue game. Just hope the error isn't discovered with 1 second left in OT and a tie score.

A scorebook error can be corrected anytime until the score is declared final by the officials. In Fed, as long as at least on of the officials is still within the visual confines of the court, it is not officially over yet. So yes, they could add the point in this sitch and play another overtime if it was simply an addition error by the scorekeeper.
This situation does not fall under the correctable error rule. This is simply a scorebook error.