AlGaN solar-blind ultraviolet (SBUV) detectors have potential application in fire monitoring, corona discharge monitoring, or biological imaging. With the promotion of application requirements, there is an urgent demand for developing a high-performance vertical detector that can work at low bias or even zero bias. In this work, we have introduced a photoconductive gain mechanism into a vertical AlGaN SBUV detector and successfully realized it in a $p\text{-}i\text{-}n$ photodiode via inserting a multiple-quantum-well (MQW) into the depletion region. The MQW plays the role of trapping holes and increasing carrier lifetime due to its strong hole confinement effect and quantum confinement Stark effect. Hence, the electrons can go through the detector multiple times, inducing unipolar carrier transport multiplication. Experimentally, an AlGaN SBUV detector with a zero-bias peak responsivity of about 0.425 A/W at 233 nm is achieved, corresponding to an external quantum efficiency of 226%, indicating the existence of internal current gain. When compared with the device without MQW structure, the gain is estimated to be about ${10}^3$ in magnitude. The investigation provides an alternative and effective approach to obtain high current gain in vertical AlGaN SBUV detectors at zero bias.