Consider a grid made up of the center points of each squares in the original grid, as shown below:

 

 

Now, each pair of adjacent squares in the original grid is corresponding to an edge in the new dashed grid. There are $$(m-1)\times n + (n-1)\times m = \boxed{2mn - m - n}$$ edges in the new grid, therefore there are this number of ways to select two adjacent squares.