Alekhine one said he probably played over 50,000 chess games.
I have to imagine he must have played more than 3 Birds.
But the following are known.
I hope the three are in the Chess Star Series on Alekhine with notes.
[Event "Stockholm sim"]
[Site "Stockholm"]
[Date "1914.??.??"]
[Round "0"]
[White "Alekhine,Alexander"]
[Black "NN"]
[Result "1-0"]
[Eco "A03"]
1.f4 d5 2.e3 Nf6 3.Nf3 Bg4 4.Be2 Nc6 5.0-0 e6 6.b3 d4 7.Bb2 dxe3 8.dxe3 Bd6
9.Nbd2 Bc5 10.Nc4 b5 11.Ba3 b4 12.Bb2 Qe7 13.Nfe5 Rd8 14.Nxc6 Rxd1 15.Bxd1 Bxe3+ 16.Nxe3 Qc5
17.Bd4 Qxc6 18.Nxg4 Nxg4 19.Bxg4 Qxc2 20.Rf2 Qe4 21.Bxg7 Rg8 22.Bf3 Qe3 23.Bf6 Qc5 24.Rd1
1-0

This one is amazing. Playing a Bird against Tarrasch in one of the greatest Tournaments. Indication is finals. Which means playing for a prize. I believe this was the tournament where the Czar gave out the first GM titles.
[Event "St Petersburg final"]
[Site "St Petersburg"]
[Date "1914.05.10"]
[Round "2"]
[White "Alekhine,Alexander"]
[Black "Tarrasch,Siegbert"]
[Result "1-0"]
[Eco "C32"]
1.f4 e5 2.e4 d5 3.exd5 e4 4.d3 Nf6 5.dxe4 Nxe4 6.Nf3 Bf5 7.Be3 c6 8.Bc4 b5
9.Bb3 c5 10.d6 c4 11.Qd5 Nd7 12.Qxf5 Nxd6 13.Qd5 Be7 14.0-0 0-0 15.Nc3 Nf6 16.Qd2 cxb3
17.axb3 b4 18.Nd5 Nf5 19.Nxe7+ Qxe7 20.Rfe1 Rfd8 21.Bd4 Nxd4 22.Nxd4 Qc5 23.Rad1 Rd5 24.h3 Rad8
25.c3 h6 26.Qd3 Qd6 27.Qf3 Nh5 28.Re4 Nf6 29.Re3 Nh5 30.Rf1 Nf6 31.Rfe1 Qc5 32.Kh2 Rc8
33.R1e2 Kf8 34.Re5 Rcd8 35.Nf5 Qb6 36.Qg3 Nh5 37.Qh4 Rxe5 38.fxe5 Rd1 39.Re3 Qg6 40.Qxb4+
1-0

[Event "Paris sim"]
[Site "Paris"]
[Date "1925.??.??"]
[Round "0"]
[White "Alekhine,Alexander"]
[Black "NN"]
[Result "1-0"]
[Eco "C00"]
1.f4 e6 2.e4 d5 3.Nc3 Ne7 4.Nf3 Ng6 5.d3 Bb4 6.Bd2 d4 7.Ne2 Bxd2+ 8.Qxd2 c5
9.b4 b6 10.bxc5 bxc5 11.g3 Nc6 12.Bg2 Rb8 13.0-0 0-0 14.Rab1 Ba6 15.h4 h6 16.h5 Nge7
17.g4 Qa5 18.Qxa5 Nxa5 19.Ne5 Rfd8 20.f5 exf5 21.exf5 f6 22.Ng6 Nxg6 23.fxg6 Bb7 24.Nf4 Bxg2
25.Kxg2 Rb6 26.Rxb6 axb6 27.Re1 Nc6 28.Re6 Nb4 29.Re7 Kf8 30.Rf7+ Kg8 31.Ne6 Re8 32.Nxg7 Re2+
33.Kf1 Rxc2 34.Nf5 Nxd3 35.Nxh6+ Kh8 36.Rh7+ 1-0

7 dxc5? Nf2! 8 Kxf2 Bxg3+ is a nice trap that's worth remembering.

Anyway, MCO-15 gives the 1 f4 e5 2 fxe5 d6 3 exd6 Bxd6 4 Nf3 Nf6 5 d4 Ng4 6 Qd3 c5 7 Nc3 cxd4 8 Qxd4 O-O 9 Bg5 as slightly better for white. White has an extra pawn, and his development is decent, but the extra pawn is isolated, Black has open lines, and he might be able to gain some tempi attacking the White queen (... Nc6 is an obvious example). It's a game.

1.f4 e5 2.fxe5 d6 3.exd6 Bxd6 4.Nf3 Nf6 5.d4 Ng4 6.Qd3 c5 7.Qe4 Be6 (this is pretty weak, even a quirky move like Kf8 is better) 8. Ng5 Nf6 9. Nxe6 Nxe4 10.Nxd8 Kxd8 11.Nd2 (there are stronger moves than this, one of them being Nc3 with a sample line 11.Nc3 Re8 12.Nxe4 Rxe4 13.dxc5 Bxc5 14.g3 Nc6 15.Bg2 Re7 16.Bf4 Nd4 17.0-0-0 Ke8 18.e4 Ne2+ 19.Kd2 Nxf4 20.gxf4 Rd8+ 21.Ke2 Red7 22.Rxd7 Rxd7 with a sizable advantage for white.) 11... Nxd2 12.dxc5 Nxf1 13.cxd6 Nxh2 14.Rxh2 Nc6 15.Kf2 Kd7 16.Bf4. A pleasant ending for white, but tenable for black.

As for the 4...g5 given line:

1.f4 e5 2.fxe5 d6 3.exd6 Bxd6 4.Nf3 g5 5.g3 g4 6.Nh4 Ne7 7.d4 Ng6 8.Nxg6 hxg6 9.Qd3 Nc6 10.c3 Bf5 11.e4 Qe7 12.Bg2 0-0-0 13.Be3 (Here a simple bishop move takes care of all of black's troubles) 13...Be6. Black is alright in all lines, just wanted to point out that the fork 14.d5 fails to 14...Rh2 15.0-0 Bxg3 16.Bg5 Rxg2+ 17.Kxg2 Qxg5 18.Qxg3 Bxd5 19.exd5 Qxd5+ 19.Kg1 Ne5 and black has a very comfortable position.

Here are a few ideas:

1 f4 e5 2 fxe5 d6 3 exd6 Bxd6 4 Nf3 Nf6 5 d4 Ng4 6 Qd3 c5 7 Qe4+ Bd6 8 Ng5 Bxh2:
A) 9 Nxe6 Qh4+ 10 Kd2 fxe6
A1) 11 Qxe6+ Kd8 12 d3 cxd4 with a lively position.
A2) 11 Rxh2 Qg5+ 12 e3 Nxh2 13 Qxe6+ Kd8 14 Qd6+ Nd7 15 Qxh2 Rf8 16 Be2 cxd4 17 Ke1 Rc8 with a lively position again.
B) 9 Rxh2 Nxh2 10 Nxe6 fxe6 11 Qxe6+ Qe7 12 Qc8+ Qd8
B1) 13 Qe6+ is a draw offer.
B2) 13 Qxb7 O-O is good for Black.
B3) 13 Qxd8+ Kxd8 and Black will have a small material advantage because 14 dxc5 runs into 14 ... Rf8. However, this small material advantage is compensated by White's passed pawns. But black will obtain a lead in development. Black's definitely not losing here, but I doubt he is winning.