See the original puzzle
The first player has the winning strategy. The first move is to place right in the middle of the table. After the second player places a quarter, the first player plays a quarter on the exact opposite side of the table. The idea is to keep the arrangement of quarters perfectly symmetric about the center of the table. No matter where the second player places a quarter, there will always be an empty spot on the exact opposite side of the table. Eventually, the second player will have no place to put a quarter.
If you want a more serious version of this game (ie, one you could actually play, because the winning strategy is not at all obvious), you could try the "misère" version. A misère game is simply a game where the win and lose conditions are switched around. So to win the misère version, you must be unable to place a quarter during your turn. As far as I know, there is no simple symmetry-based solution to the misère game.