The actual quote is generally translated as "I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use. "
While none of the many, many websites that quote it seem to deign to give a source for the passage, it’s actually legit.
The quotation is from his 1615 Letter to the Grand Duchess, i.e. Christina of Lorraine. ultrafilter’s version clearly (roughly) derives from the translation in Stillman Drake’s compilation Discoveries and Opinions of Galileo (Anchor, 1957, p183):
Given it’s general argument that such means can outweigh arguments from Scripture in certain circumstances, the Letter is particulary central to the whole subsequent Galileo affair.