squareNumberQ[{n_Integer,m_Integer}]:= IntegerQ[Sqrt[n^2+m^2]] ;
allPossiblePairs = Flatten[Table[{n,m},{n,1,10},{m,1,10}],1] ;
squareNumbers = Select[allPossiblePairs, squareNumberQ]
(* {{3,4},{4,3},{6,8},{8,6}} *)

Flatten[Table[{n,m},{n,1,10},{m,n,10}],1]

DeleteCases[squareNumbers,#1[[_,1]]==#2[[_,2]]&]