conv01 = Map["["<>StringJoin@@Riffle[#,","]<>"]"&,Map[ToString,#,{2}],{0,1}]&

list = {{1, 2, "test3"}, {3, 4, "test4"}}

"[[1, 2, "test3"], [3, 4, "test4"]]"

conv03 = Map["[" <> ToString@Row[#, ","] <> "]" &, list, {0, 1}]

"[" <> ToString@Row[{1, 2, 3, 4}, ","] <> "]"

ExportString[{
  {1, 2, "Three"},
  {4, 5, "{Six}"}
  }, "JSON"]

list = {{1, 2, "test3"}, {3, 4, "test4"}};
StringReplace[ToString@FullForm@list, "List[" -> "["]

"[[1, 2, "test3"], [3, 4, "test4"]]"

bracketFormat[expr_] :=
 StringReplace[
  ToString[expr, InputForm],
  {s : Shortest[""" ~~ __ ~~ """] :> s, "{" -> "[", "}" -> "]"}
 ]

list = {{1, 2, "test3"}, {3, 4, "test4", "trick{}"}};

bracketFormat[list]

"[[1, 2, "test3"], [3, 4, "test4", "trick{}"]]"

[[1, 2, "test3"], [3, 4, "test4", "trick{}"]]

With[{s = StringJoin@RandomChoice[CharacterRange["a", "z"], 30]},
    StringReplace[ToString[list /. {x_String :> """ ~~ x ~~ """, List -> s}], s -> ""]
]
(* [[1, 2, "test3"], [3, 4, "test4"]] *)

list={{1,2,"123"},{1,2,"123"}};

Module[{$replace$},
    StringReplace[ToString@FullForm@Map[ $replace$@@#&,list,{0,1}],ToString@$replace$->""]
]

ToExpression@"{{2,3,"List[thiswouldbreaksomeothermethods] test{123}"},{3,5,"more [stuff]"}}" /. List :> ("[" <> StringJoin[ToString /@ {##}~Riffle~","] <> "]" &)

{{2,3,"List[thiswouldbreaksomeothermethods] test{123}"},{3,5,"more [stuff]"}} /. List :> ("[" <> StringJoin[ToString /@ {##}~Riffle~","] <> "]" &)

"[[2,3,List[thiswouldbreaksomeothermethods] test{123}],[3,5,more [stuff]]]"

ListFormat -> {"[", ",", "]"}

quoted /: TextString[quoted[str_]] := """ <> str <> """;

quoted /: TextString[quoted[str_String]] := """ <> str <> """;
quoted /: TextString[quoted[str_]] := TextString@str;

TextString[MapAll[quoted, list], ListFormat -> {"[", ",", "]"}]
(* "[[1,2,"test3"],[3,4,"test4"]]" *)