Why would black queen not take white?

FAIR ENOUGH

They are all right in that.
Why black queen don't take white queen?

Why white queen do not take rook on B7?

If White plays Qxb7, then Black has plenty of options to prevent mate on the SECOND move. Moves such as ...Qd8, ...Qe8, ...Qf8, or even ...Qxc2+.

White clearly has Black in big trouble, and a mate is near-certain. His trouble, if he wants to mate in TWO, is to play a move that prevents Black going ...Qc3+ or ...Qxc2+ (or ...Rxb3+), because that would force White to make his second move with his king, and thus make it impossible to mate on the second move.

this is why White's first move must CHECK black - to prevent Black checking with his first move. Obviously Qg2 fails (after ...Kf8, there is no mating move since the white Q cannot reach the back rank except at g8 where it will be captured, nor can the white R, and White would have to continue with 2. Qg7+ Ke8 3. Qg8++, mate in three BUT NOT IN TWO.)
and obviously Qg6+, Qxh7+ and Qe8+ would be silly.

Qg4+ works because if Black moves his K (to f8 or h8), then White replies 2. Qxb8++. Black's only other move is 1... Qxg4, answered by 2. Re8++ : getting the black Q off the back rank to prevent it guarding e8 or f8 OR checking the white king, allows the rook checkmate.

Agreed with Jonellis's explanation except for:

1st paragraph: ...Qd8 move would in fact allow mate after Rxd8#.

Last paragraph, 2nd line: 1...Qxg4 is answered by 2.Rd8#, not 2.Re8, which isn't possible.

Just minor details...