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 Gödel-type spacetime in \((2+1)\)-dimensions, where such spacetime is the combination of the flat Gödel-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.

在本文中，我们分析了在\((2+1)\)维度的圆锥哥德尔型时空中具有反常磁矩的非交换量子霍尔效应的相对论和非相对论能谱（费米子朗道能级），其中这样的时空是平坦的哥德尔型时空与宇宙弦（圆锥形引力拓扑缺陷）的组合。为了分析这些能谱，我们从极坐标中具有最小和非最小耦合的非交换狄拉克方程开始。使用四分体形式，我们得到二阶微分方程。接下来，我们精确求解这个微分方程，得到广义拉盖尔方程，以及总相对论能量的二次多项式方程。通过求解这个多项式方程，我们得到了费米子和反费米子的相对论能谱。此外，我们还分析了系统的非相对论极限，得到了非相对论能谱。在这两种情况（相对论和非相对论）中，我们详细讨论了每个光谱的特征以及这些光谱中所有参数和物理量的影响。将我们的问题与其他作品进行比较，我们验证了我们的结果概括了文献中的几个特定案例。