Tolkien uses the first definition of “queer” – strange, odd, peculiar, eccentric – and mostly in the first half of the book.  Eccentricities are funny, of course.

• 01.005 got something a bit queer
• 01.023 and with the spike on his staff scratched a queer sign
• 01.093 Gets funny queer fits,
• 06.024 He gave Bilbo a queer look from under his bushy eyebrows,
• 06.086 He used to turn queer if he looked over the edge
• 06.088 He was feeling very queer indeed
• 07.092 in a queer language
• 07.116 They must have looked very queer from outside,
• 07.126 queer,
• 08.003 There were queer noises too,
• 11.028 He had a queer feeling that he was waiting for something.
• 16.018 if it is that queer little creature that is said to be their servant.’

“queer, adj.1.” OED Online. Oxford University Press, March 2015. Web. 11 May 2015.