In the subexpression d(D*(1/r^2)*(dq/dr)) is D a constant? Does the leading "d" indicate differentiation? If so, then with respect to which variable?
Does dq(0,t)/dr indicate dq/dr evaluated at (0,t) ?
You show the condition c=cs in your second boundary condition, but you had not mentioned c before. Is c a function of r and t ? Why mention the "r=R" if r is not involved in kl*(c-cs) ? At c=cs then (c-cs) would seem to be 0, and since no matter what finite value kl is, kl*0 is going to be 0, you seem to be indicating that dq/dt will be 0 whenever c=cs -- was that your intention ?
Some of the combinations I tried only have q(r,t) = 0 as a solution, and the other combinations didn't resolve. I will need to see your clarifications to go further.
Note to myself:
pdsolve([diff(q(r, t), t) = (diff(D1*(diff(q(r, t), r))/r^2, t))/r^2, q(r, 0) = 0, (D(q))(0, t) = 0], q(r, t))