This paper introduces CompressedMediQ, a novel hybrid quantum-classical machine learning pipeline specifically developed to address the computational challenges associated with high-dimensional multi-class neuroimaging data analysis. Standard neuroimaging datasets, such as large-scale MRI data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) and Neuroimaging in Frontotemporal Dementia (NIFD), present significant hurdles due to their vast size and complexity. CompressedMediQ integrates classical high-performance computing (HPC) nodes for advanced MRI pre-processing and Convolutional Neural Network (CNN)-PCA-based feature extraction and reduction, addressing the limited-qubit availability for quantum data encoding in the NISQ (Noisy Intermediate-Scale Quantum) era. This is followed by Quantum Support Vector Machine (QSVM) classification. By utilizing quantum kernel methods, the pipeline optimizes feature mapping and classification, enhancing data separability and outperforming traditional neuroimaging analysis techniques. Experimental results highlight the pipeline's superior accuracy in dementia staging, validating the practical use of quantum machine learning in clinical diagnostics. Despite the limitations of NISQ devices, this proof-of-concept demonstrates the transformative potential of quantum-enhanced learning, paving the way for scalable and precise diagnostic tools in healthcare and signal processing.