There is an interesting way to code it.

For each piece we can have a function like this:
pair<string, int> get_king(int n, int x0, int y0)
{
  int cnt = 0;
  for (int dx = -1; dx <= 1; dx++)
    for (int dy = -1; dy <= 1; dy++)
      if (dx != 0 || dy != 0)
      {
        int x = x0 + dx;
        int y = y0 + dy;
        cnt += (x >= 1 && x <= n && y >= 1 && y <= n);
      }

  return { "King", cnt };
}

In the main code we just do the following loop:
for (auto get : { get_king, get_knight, get_bishop, get_rook, get_queen })
{
  auto res = get(n, x, y);
  cout << res.first << ": " << res.second << endl;
}

After that the function for queen look very easy:
pair<string, int> get_queen(int n, int x0, int y0)
{
  int cnt = get_rook(n, x0, y0).second + get_bishop(n, x0, y0).second;
  return { "Queen", cnt };
}

If you have any ideas on how to code it better, please write them here :)