A 6*6 board will have 3 black squares and 3 white ones in each row and column.
Lets choose a random black box for the first coin and keep it fixed, we have 6*3=18 options for the first coin.
Now, we can't place the second coin on the 3 white boxes in that row and also the 3 white boxes in the corresponding column. So, for the second coin we have 18-(3+3)=12 options.
So the total number of ways: 18*12=216