You can plug in your own method into SumConvergence. Here's a start. Adapt as desired.
groupTermsCheck // ClearAll;
groupTermsCheck[nTerms_, OptionsPattern[]][expr_, k_] := Module[{j},
SumConvergence[expr, k] &&
SumConvergence[Sum[expr, {k, nTerms*j, nTerms*j + nTerms - 1}], j]
];
groupTermsCheck[expr_, k_] := groupTerms[2][expr, k];
SumConvergence[(-1)^(n)/(Sqrt[n] + (-1)^n), n,
Method -> groupTermsCheck[2]]
(* False *)
