I've had Q v Q endings a few times. Loads were agreed drawn.
This is the final position from G.Chandler - O.Penrose 2002 (Oliver is the brother of Jonathan Penrose.)
He has the big passed pawn but I was checking him all the time. We were in an allegro finish so called it a draw.
This is I. Morrison - G. Chandler Edinburgh League 1992
I get a passed pawn and shepherd it home with the Queen. Just click on move 44. It all starts from there though you may find my note after my 25th move interesting and see the whole plan come together.
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PGN
1. e4 d5 2. exd5 Nf6 3. d4 g6 4. Nf3 Bg7 5. Bd3 Nxd5 6. c3 O-O 7. O-O Nc6 8. Re1 Bg4 9. h3 Bxf3 10. Qxf3 Nb6 11. Bf4 e6 12. Bc2 Nd5 13. Na3 Nxf4 14. Qxf4 Ne7 15. Rad1 Nd5 16. Qf3 c6 17. Nc4 Qc7 18. a3 c5 19. dxc5 Qxc5 20. Ne3 Nf6 21. Qe2 Rfd8 22. c4 a5 23. Rxd8+ Rxd8 24. Rd1 Rxd1+ 25. Qxd1 Bh6 {As always playing to win. I could see no way of getting an opposite coloured Bishop attack so I to get rid of the opposite coloured Bishops and weaken g3 for a Knight.} 26. Qd3 Bxe3 27. fxe3 Qg5 28. b4 axb4 29. axb4 Nh5 30. Qe2 Ng3 {See the previous note.} 31. Qf3 h5 32. Kf2 h4 33. Qf4 Qh5 34. Qf3 Qe5 35. Qd1 Qf6+ 36. Kg1 Qc3 37. Bd3 Qxb4 38. Qc1 Qc5 39. Kf2 b5 40. Qc2 Nf5 41. Qe2 bxc4 42. Bxc4 Nxe3 43. Qxe3 Qxc4 44. Qb6 Qf4+ 45. Ke2 Qf6 46. Ke1 Kg7 47. Qe3 e5 48. Qe4 Qf4 49. Qd5 Qd4 50. Qf3 e4 51. Qg4 Qc3+ 52. Kf1 Qd3+ 53. Kg1 e3 54. Qf3 Qb1+ 55. Kh2 Qe1 56. Qe4 Qf2 57. Qd4+ Kh7 58. Qa7 Qg3+ 59. Kh1 Qf4 60. Qe7 Kg7 61. Kg1 Qf2+ 62. Kh2 e2 {My opponent resigned - he should tried....} 63. Qe8 e1=Q 64. Qh8+ Kxh8 {Stalemate. I was looking out for it so would have played 63.Qg3+ first.} *