Brain tumors are abnormal cell masses in the brain. For early-stage tumor classification based on neuroimaging techniques, particularly MRI, DL approaches such as DNNs and CNNs are frequently employed and have achieved moderate success. Recently, to address limitations of classical AI—such as data-driven challenges (e.g., high complexity, correlations) and limited computational resources (CPU, GPU)—quantum computing-based AI approaches have been developed, leveraging quantum mechanics principles and properties of quantum particles. This study proposes a hybrid quantum–classical integrated neural network (HQCINN) model for multi-class brain tumor classification, comparing its performance with two DNN and CNN models, with trainable parameters controlled at a similar level. The HQCINN model consists of quantum and classical components: the quantum part incorporates amplitude encoding, a multi-layer parameterized quantum circuit, and measurement operations, while the classical part includes the softmax function, loss computation, and optimization steps. Furthermore, the proposed quantum model was executed on the default.qubit state-vector simulator provided by PennyLane 0.35.1, whereas Keras was used for designing the DL models. The HQCINN model demonstrated the highest performance in distinguishing four different brain tumor types, with training/validation losses of 0.24/0.23 and accuracies of 0.91/0.92. For the DNN model, the losses were 1.31/1.03 and accuracies 0.47/0.51, while for the CNN model, losses were 0.42/0.80 and accuracies 0.78/0.65. The total execution time for HQCINN was 16 hours longer than that of the CNN. In conclusion, model selection should be scenario-dependent: HQCINN offers superior performance on complex datasets, whereas classical CNNs remain more efficient when speed is prioritized.