This paper presents a hybrid quantum-classical framework for real-time estimation of cognitive engagement from multichannel electroencephalography (EEG) using a new operational indicator called Computational Mental Energy (CME). The proposed approach integrates signal preprocessing (windowing, filtering, spectral feature extraction), spectral feature extraction, a 4-qubit variational quantum classifier (VQC) for flow-state probability estimation, and a metaheuristic optimization loop for balancing predictive quality and quantum resource cost. CME is defined as a window-level function of aggregated spectral energy, task complexity, and estimated flow probability, measured in a dedicated signal-energy unit called Vernik (Vn), with session-level aggregation rules. The system supports quantum-only, classical-only, and hybrid inference modes and is designed for streaming deployment with wearable EEG devices and server-side inference services. A single-subject pilot study involving eight cognitive activities and EEG recordings from a Muse Athena headband demonstrates that the hybrid mode (μ = 0.6) achieves 0.914 AUROC for flow-state detection, compared to 0.548 for the standalone quantum model, while reducing prediction variance by 40.9%. Validation on the IBM Marrakesh 156-qubit Heron r2 quantum processor shows strong agreement between simulator and hardware results (r = 0.869, MAE = 0.045), confirming the practical feasibility of execution on current quantum hardware. Across activities, CME rates differed significantly, with approximately a nine fold gap between coding and resting states, illustrating the framework’s ability to capture activity-dependent cognitive demand. The proposed architecture provides a reproducible pipeline for EEG-based cognitive-state analytics, resource-aware quantum inference, and future adaptive human-computer interaction systems.