There is a letter sent to the editor of the New England Journal of Medicine from 1974 that addresses this. The letter's author, Dr. Alsop, uses a recording of water to simulate the sound of water. The participants were "postoperative patients demonstrating failure with routine methods" to induce urination. They found that "
results in 80 patients revealed a satisfactory response in 60, many of whom required 20 to 30 minutes of auditory stimulation.
This is promising, but a letter to the editor is not quite peer-reviewed work.
A similar study was carried out in 2015 that found an average increase in mean peak flow rate among 18 adult males with lower urinary tract symptoms when they were exposed to the sound of running water (SRW). This does not imply that it causes an urge to urinate, but implies a possible connection between the two. The authors hypothesize that the results may be because
SRW might enhance the parasympathetic tone, which powers the detrusor muscle, and relax the resistant tone of the urethral sphincter, thus resulting in the increased PFR. Second, SRW might lead to increased physical and emotional relaxation. Additionally, feeling the urge to urinate with SRW appears
to be in line with a conditioned response in Pavlovian conditioning