In[240]:= n = 3;
m = 6;

In[242]:=
GetValue[i_, j_] :=
  If[i == 0 || j == 0 || i == n + 1 || j == m + 1, 0, matrix[[i, j]]];
GetNeighbours[i_, j_] :=
  GetValue[i - 1, j] + GetValue[i, j - 1] + GetValue[i + 1, j] +
   GetValue[i, j + 1];

In[244]:=
recur[hashMatrix_] := Block[{i, j, hash = hashMatrix, isEnd = 1},
   For[i = 1, i <= n, ++i,
    For[j = 1, j <= m, ++j,
      If[matrix[[i, j]] != 1 && GetNeighbours[i, j] == 1,
        isEnd = 0;
        matrix[[i, j]] = 1;
        hash = BitSet[hash, i*m + j - (m + 1)];
        If[mHash[[hash]] == 0,
         recur[hash];
         ];
        matrix[[i, j]] = 0;
        hash = BitClear[hash, i*m + j - (m + 1)];
        ];
      ];
    ];
   If[isEnd == 1,(*If[count<100,Print[matrix//TableForm]];*)++count];
   mHash[[hash]] = 1;
   ];

In[245]:= matrix = Table[0, {i, 1, 3}, {j, 1, 6}];
mHash = Table[0, 2^(m*n)];
count = 0;
hash = 0;
TimeConstrained[For[i = 1, i <= n, ++i,
    For[j = 1, j <= m, ++j,
      matrix[[i, j]] = 1;
      hash = BitSet[hash, i*m + j - (m + 1)];
      recur[hash];
      matrix[[i, j]] = 0;
      hash = BitClear[hash, i*m + j - (m + 1)];
      ];
    ];
  , 30];
count

Out[250]= 452