Is this quote about music correct?

There's a (famous according to Google) quote by the German-Jewish poet Berthold Auerbach:

Music is a universal language, and needs not be translated. With it soul speaks to soul.

Is the quote grammatically correct?

The quote is not ungrammatical. It does, however, some some archaic constructions which are no longer heard in modern English.

The main stickler is this bit:

Music... needs not be translated.

The phrasing "needs not" is no longer used in contemporary English, but in a literary register it is an acceptable variant for "does not need".

An additional wrinkle is that many speakers prefer to drop the -s when using this construction, treating the verb need as a modal auxiliary. This would give:

Music... need not be translated.

The latter is somewhat more common, but both constructions are well-attested. For a bonus, here is the N-gram chart comparing "need not" with "needs not" since 1830:

http://books.google.com/ngrams/graph?content=needs+not+be%2C+need+not+be&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=

As the chart shows, in the early 1800's both variants are reasonably common, though even then the variant with needs is much less popular. As time goes on the needs variant becomes increasingly less common, because do-support for negation becomes more and more exceptionless. The need variant survives somewhat better due to its analysis as a modal auxiliary.