  In this paper, we analyze the relativistic and nonrelativistic energy spectra
(fermionic Landau levels) for the noncommutative quantum Hall effect with
anomalous magnetic moment in the conical G\"odel-type spacetime in
(2+1)-dimensions, where such spacetime is the combination of the flat
G\"odel-type spacetime with a cosmic string (conical gravitational topological
defect). To analyze these energy spectra, we start from the noncommutative
Dirac equation with minimal and nonminimal couplings in polar coordinates.
Using the tetrads formalism, we obtain a second-order differential equation.
Next, we solve exactly this differential equation, where we obtain a
generalized Laguerre equation, and also a quadratic polynomial equation for the
total relativistic energy. By solving this polynomial equation, we obtain the
relativistic energy spectrum of the fermion and antifermion. Besides, we also
analyze the nonrelativistic limit of the system, where we obtain the
nonrelativistic energy spectrum. In both cases (relativistic and
nonrelativistic), we discuss in detail the characteristics of each spectrum as
well as the influence of all parameters and physical quantities in such
spectra. Comparing our problem with other works, we verified that our results
generalize several particular cases in the literature.
