  We construct T^2-symmetric charged AdS black holes in conformal gravity. The
most general solution up to an overall conformal factor contains three
non-trivial parameters: the mass, electric charge and a quantity that can be
identified as the massive spin-2 hair. We study the Dirac equation for the
charged massless spinor in this background. The equation can be solved in terms
of the general Heun's function for generic frequency \omega and wave number k.
This allows us to obtain the analytic Green's function G(\omega, k) for both
extremal and non-extremal black holes. For some special choice of back hole
parameters, we find that the Green's function reduces to simpler hypergeometric
or confluent hypergeometric functions. We study the Fermi surfaces associated
with the poles of the Green's function with vanishing \omega. We find examples
where the Fermi surfaces for non-Fermi liquids as well as the characteristic
Fermi ones can arise. We illustrate the non-trivial differences in the Green's
function and Fermi surfaces between the extremal and non-extremal black holes.
