This version gives a number:
\[Mu]0 = 4*\[Pi]*10^-7;
\[Rho]max = 0.22;
aa = 0.130;
Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]*\[Pi]*\[Rho])/\[Rho]max]);
dSq[\[Rho]_,
i_] := (Indexed[q,
i]*\[Pi]/\[Rho]max)*(Cos[(Indexed[q,
i]*\[Pi]*\[Rho])/\[Rho]max]);
\[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa];
pn[j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]],
20]);
f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]);
Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] +
m)!)*(1/(f[\[Rho]])^(Indexed[n, j] + 2));
pn1[\[Alpha]_,
j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]],
20]);
Cn\[Rho][
pn_, \[Alpha]_] := (m*(pn[\[Alpha], j]/Sin[\[Alpha][\[Rho]]]));
Cnm\[Psi][j_, pn1_,
pn_, \[Alpha]_] := ((1/
Sin[\[Alpha][\[Rho]]])*((Indexed[n, j] + 1)*
pn[\[Alpha], j] - (Indexed[n, j] - m + 1)*
Cos[\[Alpha][\[Rho]]]*pn1[\[Alpha], j]));
With[{q = {1}, i = 1, j = 1, n = {1}, m = 1, k = 1},
NIntegrate[(\[Rho]*
Cnm[j,
f] (Cn\[Rho][pn, \[Alpha]]*(k/\[Rho])*Sq[\[Rho], i] -
Cnm\[Psi][j, pn1, pn, \[Alpha]]*
D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]]
but it may not be what you are after. Your code is very difficult to read because you use the same symbols for both constants and variables, values and functions.