A water-lubricated rubber bearing (WLRB) is prone to generate frictional vibration noise under special operating conditions, which seriously affects the acoustic stealth performance of warships and threatens their navigation safety. Meanwhile, the main factor affecting the frictional vibration behavior of a WLRB is the materials of the friction pair. Therefore, this work selects a friction pair composed of a copper ring and a rubber block as the research object and studies the frictional vibration behavior of the ring–block friction pair under low-speed and starting conditions. The real friction coefficient curve is used to establish a transient dynamic finite element analysis model for the ring–block friction pair. The effects of the load, friction coefficient, and Young’s modulus on the frictional vibration behavior under special operating conditions are studied. The analysis’s results show that the frequency of the medium-high frequency friction-induced vibration disappears under low-speed operating conditions when the friction coefficient is below 0.1. During the startup process, even if the friction coefficient is very low, the medium-high frequency friction-induced vibration still exists. The research results provide ideas for future theoretical research and guidance suggestions for engineering practice.