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.