Can the White king move to g2 to checkmate?
8/8/8/8/4Q3/7k/5K2/8 w - - 0 1 1. Kg2
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Yes, this is illegal, as the two kings can not be within attacking distance of one another. All though, black is so very screwed in this game. A legal checkmate in 2 moves is, stalling once with the move queen to a4. The only move the king can make is h2, which will result in checkmate with the queen to h4.
Think of it this way. The goal of the game is to capture the king. If you put the king in check, it can instead just take your king and win there! Nobody cares what happens after your king falls. Pedantically though, I think any move that brings about a check to your own king is illegal so yep. Illegal!