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:

    s = StringTemplate["\!\(\*SuperscriptBox[\(``\), \(2\)]\)"];
    func = Function[x, 
      StringJoin @@ Riffle[TemplateApply[s, #] & /@ x, "+"]];
    Grid[MapThread[{#1, "=", #2, "=", ToExpression[#2]} &, {func /@ trip, 
       func /@ ftrip}]]

yields:

![enter image description here][1]


  [1]: https://i.sstatic.net/CxI6I.png