You can use this pattern
/<div class="quote">(.*?)<\/div>/is
However, this will get tricky with nested quotes. Observing the forum thread HTML code, It looks like there is always a <br> after the div.quote element, so this may achieve the desired result for nested quotes.
/<div class="quote">(.*?)<\/div><br/is
It may not work 100% of the time though, so you should always check your results.