  In this paper we obtain exact asymptotically anti-de Sitter black hole
solutions and asymptotically Lifshitz black hole solutions with dynamical
exponents $z=0$ and $z=4$ of four-dimensional conformal gravity coupled with a
self-interacting conformally invariant scalar field. Then, we compute their
thermodynamical quantities, such as the mass, the Wald entropy and the Hawking
temperature. The mass expression is obtained by using the generalized off-shell
Noether potential formulation. It is found that the anti-de Sitter black holes
as well as the Lifshitz black holes with $z=0$ have zero mass and zero entropy,
although they have non-zero temperature. A similar behavior has been observed
in previous works, where the integration constant is not associated with a
conserved charge, and it can be interpreted as a kind of gravitational hair. On
the other hand, the Lifshitz black holes with dynamical exponent $z=4$ have
non-zero conserved charges, and the first law of black hole thermodynamics
holds. Also, we analyze the horizon thermodynamics for the Lifshitz black holes
with $z=4$, and we show that the first law of black hole thermodynamics arises
from the field equations evaluated on the horizon. Furthermore, we study the
propagation of a conformally coupled scalar field on these backgrounds and we
find the quasinormal modes analytically in several cases. We find that for
anti-de Sitter black holes and Lifshitz black holes with $z=4$, there is a
continuous spectrum of frequencies for Dirichlet boundary condition; however,
we show that discrete sets of well defined quasinormal frequencies can be
obtained by considering Neumann boundary conditions.
