In Fahrenheit 451, what are Montag and Beatty arguing about in the dream Beatty tells about?

You are referring to the second section of the book "The
Sieve and the Sand." This is after Montag has spoken with Faber and is actually wearing
an ear piece so that Faber can feed Montag information and hear what is going
on.

Beatty laughs and casually comments that he had a dream
that him and Montag were arguing spitting back and forth quotes from famous authors,
philosophers and doctors. The root of it is the conflict that Montag and Beatty would be
having if Montag were courageous to speak up his newfound beliefs about knowledge and
books.

Beatty takes this opportunity to teach Montag that
the way of the firemen is the right way. He quotes books that seem to support the values
of innocence is bliss and that books cause chaos and confusion rather than knowledge and
order.

Beatty tells Montag that in the dream he used the
words of the book to refute all of Montag’s arguments. He gleefully announces “what
traitors books can be! You think they’re backing you up, and they turn on you” In the
end of the dream, Montag and Beatty get into the Salamander (the firetruck) in happy
silence and amicably drive back to the firehouse in
peace.

Beatty tells Montag that in the dream he used the
words of the book to refute all of Montag’s arguments. He gleefully announces “what
traitors books can be! You think they’re backing you up, and they turn on you." In the
end of the dream, Montag and Beatty get into the Salamander (the firetruck) in happy
silence and amicably drive back to the firehouse in
peace.

This exchange shows that Beatty knows that Montag is
wavering about what is right and wrong in regard to books and knowlege. It also seems
that Beatty concocts this dream to bring up the topic without accusing Montag outright
of these traitorous thoughts.