I think that last subs command is getting confused because q is defined as the solution of the ode. I'm not sure why that's a problem (or if I'm even correct about that), but you can get around this by using a different variable name as the solution of dsolve:
syms q(t) Q(s) P_c Q_c
cond_q = q(0) == 0;
ode_capacitor = diff(q(t),t) + P_c*q == Q_c;
qsol(t) = dsolve(ode_capacitor,cond_q);
i_c(t) = diff(qsol(t),t);
I_c(s) = laplace(i_c(t));
% Laplace transform of the ode
E(s) = laplace(ode_capacitor);
E(s) = subs(E(s),laplace(q(t),t,s),Q(s));
E(s) = isolate(E(s),Q(s));
Q(s) = rhs(E(s));
Q(s) = subs(Q(s),q(0),0);
% compare Q(s) with laplace(qsol(t))
[Q(s) simplify(laplace(qsol(t)))]
% verify relationship between charge and current
[s*Q(s) I_c(s)]
ans =
[ Q_c/(s*(P_c + s)), Q_c/(s*(P_c + s))]
ans =
[ Q_c/(P_c + s), Q_c/(P_c + s)]
