My copy of thermal Physics, by Kittel and Kroemer, mentions this
quote in a homework problems designed to show that the claim
"six monkeys, set to strum unintelligently on typewriters for millions
of years, would be bound in time to write all the books in the
British Museum" is really nonsense. (Kittel and Kroemer, 2nd ed., p. 53).
They list their source for the quote as James Jeans' book Mysterious
Universe, Cambridge University Press, 1930, p.4. They say that Jeans
attributed the quote to Huxley, which may be the origin of this connection
to Huxley. Since I don't have Jeans' book, I can't tell you exactly
what he said.
Please let us all know what you find out. I have a similar story
to report about a famous alleged quotation.