use Tuple class, declare a shift array - very organized.

ConnectedCellInaGrid4.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ConnectedCell
{
    using System.Reflection;

    class Program
    {
        static int[] shifts = new int[] { -1, 0, 1 };

        static void Main(string[] args)
        {
            int m, n;
            m = int.Parse(Console.ReadLine());
            n = int.Parse(Console.ReadLine());
            int[,] table = new int[m, n];

            for (int i = 0; i < m; i++)
            {
                var str = Console.ReadLine().Split(' ');
                for (int j = 0; j < n; j++)
                {
                    table[i, j] = int.Parse(str[j]);
                }
            }

            int max = 0;
            for (int i = 0; i < m; i++)
            {
                for (int j = 0; j < n; j++)
                {
                    if (table[i, j] == 1)
                    {
                        max = Math.Max(RegionWeight(table, i, j, m, n), max);
                    }
                }
            }

            Console.WriteLine(max);
        }

        private static int RegionWeight(int[,] table, int i, int j, int m, int n)
        {
            int count = 1;
            table[i, j] = 0;
            var coords = new List<Tuple<int, int>>();
            coords.Add(new Tuple<int, int>(i, j));

            while (coords.Count > 0)
            {
                var tmp = new List<Tuple<int, int>>();
                foreach (var coord in coords)
                {

                    for (int shiftI = 0; shiftI < shifts.Length; shiftI++)
                    {
                        if (InBounds(coord.Item1 + shifts[shiftI], m))
                        {
                            for (int shiftJ = 0; shiftJ < shifts.Length; shiftJ++)
                            {
                                if (InBounds(coord.Item2 + shifts[shiftJ], n)
                                    && table[coord.Item1 + shifts[shiftI], coord.Item2 + shifts[shiftJ]] == 1)
                                {
                                    tmp.Add(new Tuple<int, int>(coord.Item1 + shifts[shiftI], coord.Item2 + shifts[shiftJ]));
                                    table[coord.Item1 + shifts[shiftI], coord.Item2 + shifts[shiftJ]] = 0;
                                }
                            }
                        }
                    }
                }

                count += tmp.Count;
                coords = tmp;
            }

            return count;
        }

        private static bool InBounds(int x, int max)
        {
            return x >= 0 && x < max;
        }
    }
}