Define the F function as follows

Define the F function as follows

F[m_,i_,j_]:=Module[{rls},
  rls = {
    f[arg1_, arg2_] :> f[arg1 - 2, arg2] /; MemberQ[{i, j}, arg2], 
    f[arg1_, arg2_] :> f[arg1 - 1, arg2] /; Not[MemberQ[{i, j}, arg2]]
   };

  Fold[ReplaceAll[#1, #2] &, f[m, 1] f[m, 2] f[m, 3] f[m, 4], rls]
]

Evaluating F[m,1,3] returns

f[-2 + m, 1] f[-2 + m, 3] f[-1 + m, 2] f[-1 + m, 4]

Effectively, what the solution does is to apply repeatedly the transformation rules for f[m,i]; it first tackles the case where i,j's in the f's are equal to the inputs and then deals with all the remaining cases (ie those indexes not equal to either of the inputs).