Do quotes constitute a duplicate content problem?

Using quoted text passages from other websites on your own site does not constitute a duplicate content problem and you will also not get hit by a Google Penalty because of it, provided you properly cite the source for the text in the source code.

Correctly citing quotes in the sourcecode

Quotations or quoted text passages are not identified as duplicate content. You should make sure though that the quotes are properly cited within the source code as follows:

When you are using quotes, you should make sure that the semantic markup in the sourcecode is correct:

<blockquote>Here is the quoted text <cite>Here is the name of
the quoted author or the source</cite></blockquote>

Video explanation by Matt Cutts/ Google on the subject