Multi-beam radar sensing systems are emerging as powerful tools for non-contact motion tracking and vital-sign monitoring in healthcare environments. This paper presents the design and experimental validation of a synchronized millimeter-wave multi-radar tracking system enhanced by a modified spherical gradient-index (GRIN) Luneburg lens. Five commercial FMCW radar modules operating in the 58--63 GHz band are arranged in a semi-circular configuration around the lens, whose tailored refractive-index profile accommodates bistatic radar modules with co-located transmit (TX) and receive (RX) antennas. The resulting architecture generates multiple fixed high-gain beams with improved angular resolution and minimal mutual interference. Each radar operates independently but is temporally synchronized through a centralized Python-based acquisition framework to enable parallel data collection and low-latency motion tracking. A 10-cm-diameter 3D-printed prototype demonstrates a measured gain enhancement of approximately 12 dB for each module, corresponding to a substantial improvement in detection range. Full-wave simulations and measurements confirm effective non-contact, privacy-preserving short-range human-motion detection across five 28-degree sectors, providing 140-degree total angular coverage. Fall-detection experiments further validate reliable wide-angle performance and continuous spatial tracking. The proposed system offers a compact, low-cost, and scalable platform for millimeter-wave sensing in ambient healthcare and smart-environment applications.