Gliomas, which are the most common malignant primary brain tumors, present significant challenges in terms of varying survival rates, treatment modalities, and prognostic processes between patients with low-grade gliomas (LGGs) and high-grade gliomas (HGGs). Accurate classification and grading of LGGs and HGGs are crucial for appropriate treatment planning and assessment of overall prognosis. The classification and grading of gliomas have undergone evolution over time, and the inclusion of molecular markers in the classification of glioma tumors by the WHO Central Nervous System (CNS) Tumors Classification in 2016 and the incorporation of advanced molecular diagnostics in 2021 have improved glioma tumor characterization. However, the high cost and limited accessibility of molecular genetic tests, coupled with the time-consuming nature of obtaining results, can lead to delays in critical treatment decisions. To address these challenges, various classical artificial intelligence methods, such as machine learning and deep learning, have been applied to problems in this field, yielding a certain level of success. Nevertheless, the continuous expansion of medical data dimensions, the inherent noise level in the data, and the limitations of the classical vector space pose significant challenges that classical artificial intelligence methods struggle to overcome. Recent studies have demonstrated that the use of quantum computing and quantum artificial intelligence technologies in healthcare not only addresses these problems but also accelerates complex data analyses and processes large datasets more efficiently. This paper presents a novel hybrid quantum (or classical-quantum) computing model aimed at differentiating between LGGs and HGGs using data from The Cancer Genome Atlas (TCGA). To the best of our knowledge, this study is the first to investigate the classification of LGGs and HGGs using a hybrid classical-quantum framework within the TCGA dataset. In the classical part of the study, an ensemble feature selection method was used to identify the most important molecular markers and clinical features within the TCGA glioma dataset. In the quantum section, six variational quantum classifier (VQC) models with different hyperparameters are proposed. These classifiers are subsequently used to differentiate between LGGs and HGGs using the features obtained from the ensemble model. The computational results show that among the six VQC models, the VQC-1 model, which incorporates Rx and CX gates in the feature map and Ry, Rz and CY gates in the parameterized quantum circuit and utilizes the AQCD optimization method, achieves the highest classification accuracy of 0.74. This study provides a novel perspective on the classification of glioma tumors by combining classical and quantum computing methods.