I sincerely doubt it. I expect that to use the books as a basis, one would need the approval of the Tolkien estate, and it's both simpler and more profitable to buy the rights from New Line Cinema to produce it as film merchandising. That's a pity, because Tolkien probably would have been quite happy to see trivia about his books used in a game, provided that it was accurate.
You could always make your own question cards from the books, but then you'd have to give yourself enough time to forget the answers for it to be any fun.
Incidentally, years before the films came out, I saw a 'pocket' travel version of Trivial Pursuit. As per the tradition, there was a quotation on the side of the box:
Quote:
"What has it got in its pocketses?" Gollum
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