This is not particularly fast but works:
f = (-1 + Sqrt[3 + 6 #^2])/2 &;
q = IntegerQ /@ (f /@ Range[1000000]);
answer = Pick[Range[1000000], q]
yields the triples : {1, 11, 109, 1079, 10681, 105731}
fanswer = f /@ answer
{1, 13, 133, 1321, 13081, 129493}
Dropping first case which is 0^2+1^2+2^2=1^2+2^2:
trip = Rest[{(# - 1), #, # + 1} & /@ answer];
ftrip = Rest[{#, (# + 1)} & /@ fanswer];
s = StringTemplate["\!\(\*SuperscriptBox[\(``\), \(2\)]\)"];
func = Function[x,
StringJoin @@ Riffle[TemplateApply[s, #] & /@ x, "+"]];
Grid[MapThread[{#1, "=", #2, "=", ToExpression[#2]} &, {func /@ trip,
func /@ ftrip}]]
yields:
