Maybe so:
ClearAll["Global`*"]; Remove["Global`*"];(* Clears the kernel *)
.
b[x_] := FindRoot[b == x - 1/10*Quantile[NormalDistribution[0, 1], b][[1]], {b, 1/2}]
b[0.1]
(* {b -> 0.188386} *)
b[2]
(* Give error messages because, InverseErfc[x] exist only for 0 < x < 2*)
Func[x_, ?_, ?_, s_] := FindRoot[b == x - s*Quantile[NormalDistribution[?, ?], b][[1]], {b, 1/10}]
Func[0.1, 0, 1, 1/10]
(* {b -> 0.188386} *)
Another way:
f[b_?NumericQ, ?_?NumericQ, ?_?NumericQ, s_?NumericQ] := s*Quantile[NormalDistribution[?, ?], b]
x = 0.1;
? = 0;
? = 1;
s = 1/10;
NMinimize[{1, -b + x - f[b, ?, ?, s] == 0}, b, Method -> "RandomSearch"][[2]] // Quiet
(* {b -> 0.188386 } *)