Checking whether it's a draw is fairly straightforward. All we have to do is check whether all the board's boxes are filled. We can do that by iterating through the board array:

public String play(int x, int y) {
  checkAxis(x);
  checkAxis(y);
  lastPlayer = nextPlayer();
  setBox(x, y, lastPlayer);
  if (isWin()) {
    return lastPlayer + " is the winner";
  } else if (isDraw()) {
    return "The result is draw";
  } else {
    return "No winner";
  }
}

private boolean isDraw() {
  for (int x = 0; x < SIZE; x++) {
    for (int y = 0; y < SIZE; y++) {
      if (board[x][y] == '') {
        return false;
      }
    }
  }
  return true;
}