I have my doubts, although Feynman like many others expressed similar sentiments. Dennett in the first chapter of Breaking the Spell (2006) writes:"Philosophy is questions that may never be answered. Religion is answers that may never be questioned". He attributes the quote to Anonymous, and does not mention Feynman. It appears that the quote's author, Feynman or not, prefers philosophy to religion in this allusion.
Here is what Mermin writes about another "Feynman's" famous quote:"I noticed that not a single one of the Web sites attributing the phrase to Feynman cited a source or hinted at the circumstances under which he had said it... I suspect that it is only Feynman’s habitual irreverence that has linked him in the minds of many to the phrase “shut up and calculate”... Merton has taught us that it is only among the high and mighty that people tend to look... Broaden the search to embrace the low and powerless".
As in Mermin's case not a single place sporting this quote cites anyone, let alone Feynman, or even tells an anecdote as to how it was said. They do not date too far back in time either.