Checkmate

I am studying checkmates in Basic Checkmates and more... board study Three's the Charm.  White Q starts at d3 and moves to g6, then Blk K moves to h8, white Q moves to h6.  Why not just end game with white Q in g7?  As I see it blk K will be in checkmate.

Not quite. If the white queen moves to g6, then to h6 and then to g7, the queen gets taken. 

So, if white can't checkmate the king with just his queen, what is he going to do?

That was the point of the lesson. Did you get it?

No, I don't get it.  It seems to me that if you move to g7 that the blk K has nowhere to go and the rook can't take the white queen because it can not move horizontally.

The king can take the queen if it goes to g7. The queen is not protected.

Since that doesn't work, the best is to go for the draw by perpetual check.

White is trying to keep from getting checkmated!

The only way to avoid getting checkmated is to keep doing checks with the white queen.