As the main branch of microwave photonics, radio-over-fiber technology provides high bandwidth, low-loss, and long-distance propagation capability, facilitating wide applications ranging from telecommunication to wireless networks. With ultrashort pulses as the optical carrier, huge capacity is further endowed. However, the wide bandwidth of ultrashort pulses results in the severe vulnerability of high-frequency RF signals to fiber dispersion. With a time-energy entangled biphoton source as the optical carrier combined with the single-photon detection technique, a quantum microwave photonics method in radio-over-fiber systems is proposed and demonstrated experimentally. The results show that it not only realizes unprecedented nonlocal RF signal modulation with strong resistance to the dispersion but also provides an alternative mechanism to distill the RF signal out from the dispersion effectively. Furthermore, the spurious-free dynamic ranges of the nonlocally modulated and distilled RF signals have been significantly improved. With the ultra-weak detection and high-speed processing advantages endowed by the low-timing-jitter single-photon detection, the quantum microwave photonics method opens up new possibilities in modern communication and networks.