I think the following endgame is a draw.
[FEN "7k/5BpP/3K2P1/8/8/8/8/4q3 w - - 0 1"]
My reasoning: I'm playing white. I will only move my king, always. Black's king cannot move. If the queen decide to capture any of my pieces, I will take back, so that won't work. And the queen alone cannot checkmate me, but black cannot bring a piece to help, so it looks like it is a draw.
Is there any flaw in my reasoning, and if yes, what is the flaw? Is this endgame really a draw?
Here's a simpler way to see why it's winning for black: You're right that the queen cannot checkmate the king by itself, but it can stalemate the king [*] (an example pointed out by Ionut Deaconu) and that's all we need here as that would force the bishop to move. Once the bishop moves it's clear that material loss is unavoidable as either the bishop [**] or the g-pawn will drop.
[*]: Take any queen vs king situation (relevant here as the bishop cannot move and thus unable to prevent the queen from shouldering the king to a stalemated situation).
[**]: The idea here is that with a cornered king there's no way for the bishop to move to a square guarded by its king while still protecting the g6 pawn.
Below a dummy example is showcased:
[title "Stalemate the king to win, example"]
[fen "7k/5BpP/3K2P1/8/8/8/8/4q3 w - - 0 1"]
1. Kd5 Qe3 2. Kd6 Qe4 3. Kd7 Qe5 4. Kc6 Qd4 5. Kc7 Qc5+ 6. Kd7 Qb6 7. Ke7 Qc6 8. Kd8 Qd6+ 9. Kc8 (9. Ke8 Qc7 10. Kf8 {Alternatively any bishop move drops the g6 pawn. E.g., 11. Be6 Qc6+ 12. Ke7 Qe4} Qd8+ 11. Be8 Qd6+ 12. Kf7 Qf6#) Qe7 10. Kb8 Qd7 11. Ka8 Qc7 {King stalemate reached} 12. Bg8 Qc6+ 13. Kb8 Qxg6 {The win is trivial now.}
Additional clarifications in regard to the question asked by Ionut Deaconu in comments:
Why 4.Kc6? You cannot push me to the corners after 4.Be8. If you want to try a zugzwang I can play Bf7 back.
The diagram above focused on the case where white tries to survive only by king moves (as is of particular interest to the OP). White can of course try to keep the king near the bishop on f7 and make waiting bishop moves in order to avoid their king from being cornered altogether. However, this is to no avail as the queen will first start off by disconnecting the king from the bishop and only then proceed to shouldering the king towards a corner and achieve the king-stalemate scenario. The disconnect is easily achieved by maneuvering along the dark squares to enter the 7th row and then push the king out. Illustrative variations are shown below:
[title "Variation in regard to questions in comments:"]
[fen "7k/5BpP/3K2P1/8/8/8/8/4q3 w - - 0 1"]
1. Kd7 Qe5 2. Be8 Qd5+ 3. Ke7 Qc5+ 4. Kd7 Qa7+ 5. Ke6 (5. Kd8 Qb7 6. Bf7 Qc6 7. Ke7 Qc7+ 8. Ke6 Qd8 {and we reach the wanted scenario again. For the rest see the main variation.}) (5. Kd6 Qb7 6. Bf7 Qc8 7. Ke7 Qc7+ 8. Ke6 Qd8 {Again the wanted scenario is reached, the rest is similar.}) Qc7 6. Bf7 Qd8 {The hard part, which is the disconnect, has been achieved once this scenario is reached, the rest is trivial now again as there are no more useful waiting bishop moves there to deal with. The rest of this variation may be done more optimally.} 7. Kf5 Qf6+ 8. Ke4 Qg5 9. Kd4 Qf5 10. Ke3 Qe5+ 11. Kd2 Qd4+ 12. Kc1 Qc3+ 13. Kd1 Qb2 14. Ke1 Qc2 15. Kf1 Qd2 16. Kg1 Qe2 17. Kh1 Qf2
According to the Lomonosov tablebases, it's mate in 40 for black. While the queen can't force mate by herself, she can force a zugzwang. In this case, when the bishop is forced to move to e8. Then, it's a matter of giving checks until the queen forks the king and bishop.
One of the longest variations given is as follows:
[FEN "7k/5BpP/3K2P1/8/8/8/8/4q3 w - - 0 1"]
[Result "0-1"]
1. Kd7 Qb1 2. Be8 Qb8 3. Bf7 Qb7+ 4. Kd6 Qc8 5. Kd5 Qc7 6. Kd4 Qc6 7. Ke5 Qd7 8. Ke4 Qd6 9. Ke3 Qe5+ 10. Kd3 Qf4 11. Kc3 Qe4 12. Kb3 Qd4 13. Ka3 Qd6+ 14. Kb3 Qc5 15. Kb2 Qe3 16. Kc2 Qa3 17. Kd2 Qf3 18. Kc2 Qe3 19. Kb2 Qd3 20. Ka2 Qc3 21. Kb1 Qd2 22. Bb3 Qb4 23. Ka2 Qa5+ 24. Kb2 Qb6 25. Kc3 Qxg6 26. Bc4 Qd6 27. Bd3 g5 28. Be4 Qc5+ 29. Kd3 g4 30. Bh1 Qa3+ 31. Kd2 Qb2+ 32. Kd3 Qb1+ 33. Ke3 Qxh1 34. Kf4 Qe1 35. Kxg4 Qe3 36. Kh4 Qg1 37. Kh3 Kxh7 38. Kh4 Kg6 39. Kh3 Kf5 40. Kh4 Qg4# 0-1
You're right, at the first sight it looks like a draw: Black's king is kept out of play and the Queen cannot checkmate alone. Black's only idea for a win would be to sac the Queen on h2 and try to promote the other pawn, but that's obviously impossible because of the White Bishop still in play.
However, there's a trick: White can be put in zugzwang by Black's Queen, which could force him to move the Bishop and lose material. For instance, if the Queen manages to push White's King to a8 with the Queen on c7, White need to move (and lose) the Bishop.
Bearing that in mind, White should try to keep his King on the d8, d7, or e7 squares, so that he can move the Bishop to e8 in case of zugzwang.
If black takes the bishop when it's on f7, white wins. If the king is on f7, the bishop on e8, and the queen on the a1-h8 diagonal, then black wins. The bishop has to be protecting the g6 pawn, otherwise, black takes that pawn and wins. If anything else, it's a draw.
EDIT: the starting position is in black's favor FEN: 7k/3K1BpP/6P1/8/8/8/8/4q3 w - - 0 1 PGN:[Variant "From Position"] [FEN "7k/3K1BpP/6P1/8/8/8/8/4q3 w - -"]
