Does the sound of running water make us want to pee?
It certainly can, although this is most likely due to an unconscious association from the sound of the running water; there is no known physiological (ie medical) explanation. When we hear the sound of a running tap, rainfall or a waterfall, it reminds us of the act of urination and thus makes us want to pass urine.
This type of association was shown by the famous experiments of Pavlov’s dogs. Pavlov was a scientist who gave dogs food and also rang a bell at the same time. After he’d done this several times, he just rang the bell, and the dogs salivated as they unconsciously expected food.