This paper proposes a hybrid quantum neural network (HQNN) for indoor user localization using received signal strength indicator (RSSI) values. We use publicly available RSSI datasets for indoor localization using WiFi, Bluetooth, and Zigbee to test the performance of the proposed HQNN. We also compare the performance of the HQNN with the recently proposed quantum fingerprinting-based user localization method. Our results show that the proposed HQNN performs better than the quantum fingerprinting algorithm since the HQNN has trainable parameters in the quantum circuits, whereas the quantum fingerprinting algorithm uses a fixed quantum circuit to calculate the similarity between the test data point and the fingerprint dataset. Unlike prior works, we also test the performance of the HQNN and quantum fingerprint algorithm on a real IBM quantum computer using cloud quantum computing services. Therefore, this paper examines the performance of the HQNN on noisy intermediate scale (NISQ) quantum devices using real-world RSSI localization datasets. The novelty of our approach lies in the use of simple feature maps and ansatz with fewer neurons, alongside testing on actual quantum hardware using real-world data, demonstrating practical applicability in real-world scenarios.