  In this paper, we analyze Dirac fermions in G\"odel-type background
space-times with torsion. We also consider the G\"odel-type spacetimes embedded
in a topological defect background. We show that relativistic bound states
solutions to the Dirac equation can be obtained by dealing with three cases of
the G\"odel-type solutions with torsion, where a cosmic string passes through
these three cases of the space-time. We obtain the relativistic energy levels
for all cases of the G\"odel-type solutions with torsion with a cosmic string,
where we show that there exists an analogy with the Landau levels for Dirac
particles. We also show that the presence of torsion in the space-time yields
new contributions to the relativistic spectrum of energies and that the
presence of the topological defect modifies the degeneracy of the relativistic
energy levels.
