1.Qf7 Qc6+ 
2.f3 Qc2+ 
3.Kxg3 Checks runs out, so Black has to move the bishop away, but doing so will mean, White will queen by Qf8+ followed by Qf5+ and c8=Q. After queening, Black can no longer do Qc2+.

1. Qa1+ Be5 
2. Qa8+ Kg7/h7 
3. c8Q 
and black has no checks on the diagonal.

I support the idea by EgoistPaul: 
1. Qf7! 
then 
1...Bf4 or 1...Be5 allows promoting by check on the 8th rank. 
 
if instead 1...Qc6+ white can simply take on g3. 
If at any moment black plays Qa8 to protect the back rank, white can check and protect c8, for example 
1. Qf7 Qc6+ 
2. Kxg3 Qa8 
3. Qf6+ Kg8 
4. Qe6+ promoting in the next one 
or 3... Kh7 
4. Qf5+

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