I do not understand why do you think that this move is hard.
Here is movei on a slow machine
It likes Qxg7 very fast in less than a second at depth 8.
I think that the position is not hard and the question if to play Qxg7 is mainly a positional question.
The main problem is that some programs do not have evaluation that tells them that white is better and for some reason they are happy with a draw score but even in case of toga1.2.1a that is not very good in test positions it can see Qxg7 in less than a minute on a slow machine.
edit:after some analysis it is clear to me that Qxg7 is equivalent to Qh4+ Ke8 Qh8+ Ke7 Qxg7 so the position is a bad test position.
New game
r1b5/ppq1kppQ/n1prp3/8/2P5/8/PP2BPPP/3R1RK1 w - - 0 1
Analysis by Movei00_8_403:
1.Qxg7
µ (-1.29) Depth: 1 00:00:00
1.Qxg7
µ (-0.99) Depth: 2 00:00:00
1.Qxg7 Rd7
³ (-0.49) Depth: 2 00:00:00
1.Qxg7 Rd7
³ (-0.49) Depth: 2 00:00:00
1.Qxg7 Rd7
= (-0.19) Depth: 3 00:00:00
1.Qxg7 Nb4 2.Bh5
= (-0.16) Depth: 3 00:00:00
1.Qxg7 Nb4 2.Bh5
= (-0.16) Depth: 3 00:00:00
1.Qh4+ Ke8
³ (-0.33) Depth: 4 00:00:00
1.Qh4+ Ke8
³ (-0.33) Depth: 4 00:00:00
1.Qh4+ Ke8
³ (-0.33) Depth: 4 00:00:00
1.Qh4+ Ke8
³ (-0.33) Depth: 4 00:00:00
1.Qxg7 Rxd1 2.Qg5+ Kf8 3.Rxd1
³ (-0.63) Depth: 4 00:00:00
1.Qxg7 Rxd1 2.Qg5+ Kf8 3.Rxd1
³ (-0.63) Depth: 4 00:00:00
1.Qxg7 Rxd1
µ (-0.93) Depth: 5 00:00:00
1.Qxg7 Rxd1 2.Rxd1 Qf4 3.Qh8
µ (-1.00) Depth: 5 00:00:00
1.Qh4+ f6
µ (-0.99) Depth: 5 00:00:00
1.Qh4+ f6
³ (-0.70) Depth: 5 00:00:00
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 5 00:00:00
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 5 00:00:00
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 6 00:00:00 23kN
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 6 00:00:00 26kN
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 7 00:00:00 38kN
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 7 00:00:00 45kN
1.Qh4+ f6 2.Qh8 Kf7 3.Bh5+ g6 4.Qh7+ Kf8 5.Qh8+ Kf7
= (0.00) Depth: 8 00:00:00 72kN
1.Qxg7 Rxd1
= (0.01) Depth: 8 00:00:00 133kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Ke7 4.Qh4+ Kf8 5.Qh8+ Ke7 6.Rxd7+ Qxd7 7.Rd1 Qc7
² (0.26) Depth: 8 00:00:00 215kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Ke7 4.Qh4+ Kf8 5.Qh8+ Ke7 6.Rxd7+ Qxd7 7.Rd1 Qc7
² (0.26) Depth: 8 00:00:01 224kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Kg8 4.Bd3 Nb4 5.Bh7+ Kh8 6.Be4+ Kg8 7.Rfe1 Rxd1 8.Rxd1
= (0.17) Depth: 9 00:00:01 384kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Kg8 4.Bd3 Nb4 5.Bh7+ Kh8 6.Be4+ Kg8 7.Rfe1 Rxd1 8.Rxd1
= (0.17) Depth: 9 00:00:02 501kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Kg8 4.Bd3 Nb4 5.Bh7+ Kh8 6.Be4+ Kg8 7.Rxd7 Qxd7 8.Re1
² (0.26) Depth: 10 00:00:03 909kN
1.Qxg7 Rd7 2.Qg5+ Kf8 3.Qh6+ Kg8 4.Bd3 Nb4 5.Bh7+ Kh8 6.Be4+ Kg8 7.Rxd7 Qxd7 8.Re1
² (0.26) Depth: 10 00:00:04 1012kN
1.Qxg7 Rd7
² (0.56) Depth: 11 00:00:12 3193kN
1.Qxg7 e5 2.Qxe5+ Be6 3.Qg5+ Kf8 4.Qh6+ Kg8 5.Rd3 Rxd3 6.Bxd3 Re8 7.Bh7+ Kh8 8.Bf5+ Kg8 9.Re1
² (0.35) Depth: 11 00:00:21 5504kN
1.Qxg7 e5 2.Qxe5+ Be6 3.Qg5+ Kf8 4.Qh6+ Kg8 5.Rd3 Rxd3 6.Bxd3 Re8 7.Bh7+ Kh8 8.Bf5+ Kg8 9.Re1
² (0.35) Depth: 11 00:00:21 5773kN
1.Qxg7 e5
² (0.65) Depth: 12 00:00:27 7163kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Qd6+ Kf7 8.Bh5 Qxh5 9.Rxa8
² (0.67) Depth: 12 00:00:30 8163kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Qd6+ Kf7 8.Bh5 Qxh5 9.Rxa8
² (0.67) Depth: 12 00:00:32 8817kN
1.Qxg7 e5
± (0.97) Depth: 13 00:00:49 13071kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Rd6+ Kf5 8.Rxf6+ Qxf6 9.Qxa8 e4 10.Qc8+ Kf4 11.Qxh3 Qxb2
± (1.15) Depth: 13 00:01:07 17840kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Rd6+ Kf5 8.Rxf6+ Qxf6 9.Qxa8 e4 10.Qc8+ Kf4 11.Qxh3 Qxb2
± (1.15) Depth: 13 00:01:10 18859kN
1.Qxg7 e5
± (0.85) Depth: 14 00:01:24 22652kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Rd6+ Kf5 8.Rxf6+ Qxf6 9.Qxa8 Ke4 10.gxh3 Qg5+ 11.Kf1 Qc1+ 12.Kg2 Qxb2
² (0.69) Depth: 14 00:01:43 28333kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Rd6+ Kf5 8.Rxf6+ Qxf6 9.Qxa8 Ke4 10.gxh3 Qg5+ 11.Kf1 Qc1+ 12.Kg2 Qxb2
² (0.69) Depth: 14 00:01:59 33373kN
1.Qxg7 e5
± (0.99) Depth: 15 00:03:16 53840kN
1.Qxg7 e5 2.Rxd6 Qxd6 3.Rd1 Qg6 4.Qh8 f6 5.Rd8 Bh3 6.Qf8+ Ke6 7.Qd6+ Kf7 8.Bh5 Qxh5 9.Rxa8 Be6 10.Rxa7
± (0.97) Depth: 15 00:04:06 68060kN
(Uri, MyTown 20.04.2007)