If the condition is that the coin touches at least one other square besides the one the center of the coin lies on, then:
Area of one square = 1.000
Area of a R0.25 radius circle centered on the coin = 0.1963
(The coin's center must stay within this circle for the coin to stay within one square.)
So the probability would be 1 - 0.1963 = 0.8037.
It seems that would hold for any size board.
If the condition is that it only touch a part of one other square, as opposed to covering a "four corners," I'd have to cipher on that some more.