Define that if a stone has a position $(x,y)$ then it is $x$ more steps upward and $y$ more steps rightward before it reaches the upper-right position. This way, we can imagine the grid to be rotated; it starts on the lattice point $(m-1,n-1)$ and players move it leftward or downward. If the coordinates are the same, then the player to move must make them not equal. If the coordinates are not the same, then the player to move can make them equal. Since $(0,0)$ has equal coordinates, then all $(x,y)$ where $x \neq y$ wins and all $(x,x)$ loses. Hence the first player wins iff $m \neq n$.

This is clearly the same as a nim Game with two heaps of sizes $n$ and $m$.

good problem

This game is equivalent to Nim$(m,n)$.The Grundy Number of $(m,n)$ is $m$ xor $n$. First player wins$\iff$Grundy Number is not $0\iff m$ xor $n \neq 0\iff m\neq n$