Are there any opening traps where your opponent can blunder into a windmill?
I did some deep, deep searching and found few results. It seems that there are no actual windmill traps that are quick and widely known. This is unless you somehow manage to run through long and arbitrary sequences that are extremely unlikely to occur in actual play. Even then, it is a matter of your opponent who actually falls for it.
If you still consider it a trap though, the one genuine such trap that I have discovered so far occurs in the Russian Game, Damiano Variation.
[FEN ""] [startply "26"] 1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4 4. Qe2 Qe7 5. Qxe4 d6 6. d4 dxe5 7. Qxe5 Qxe5+ 8. dxe5 Nc6 9. Bb5 Bd7 10. Bxc6 Bxc6 11. O-O O-O-O 12. Bf4 g5 13. Bxg5 Rg8 14. Bxd8 Rxg2+ 15. Kh1 Rxf2+ 16. Kg1 Rg2+ 17. Kh1 Rg3+ 18. Rf3 Bxf3#
The trap occurs when Black plays 13... Rg8. If White strikes the false prey with the move 14. Bxd8?, thinking they have won an exchange, then Black can play 14... Rxg2! and commence a mate in 4 using the windmill tactic.
Another alternative, as @supercat suggested in a comment, is that Black could troll some and have fun demonstrating the power of the windmill.
[FEN "2kB1b2/ppp2p1p/2b5/4P3/8/8/PPP2PrP/RN3R1K b - - 0 1"] 1... Rxf2+ 2. Kg1 Rg2+ 3. Kh1 Rxc2+ 4. Kg1 Rg2+ 5. Kh1 Rxb2+ 6. Kg1 Rg2+ 7. Kh1 Rxa2+ 8. Rf3 Bxf3+ 9. Kg1 Bc5+ 10. Kf1 Rxh2 11. Ke1 Be3 12. Nd2 Bxd2+ 13. Kf1 Be3 14. Ke1 Rh1#