It makes things easier for the reader when you are more specific about what variables a function is dependent upon.
A solution is as follows:
C(t, z) = _C1+_C2*exp((1+b)*t)
q(t) = -_C2*exp((1+b)*t)+b*_C1
Notice that once again your system is independent of z, which is most easily resolvable by making d2C/dz2 zero... but there are probably other solutions.
This is the third time that you have presented a pde that was effectively independent of one of its nominal derivatives. It would help you to learn to recognize and deal with those situations.
