Multiple realities(covers information from several alternate timelines)
A nosebleed was a condition where the capillaries in one's nose burst, causing blood loss. It could be caused by an actual blow to the nose or manifest as a psychosomatic response.
In 2255 of the alternate reality, James T. Kirk suffered a nosebleed after getting into a fight with Hendorff and other Starfleet Academy cadets at the Shipyard Bar. Later, he used wadded-up napkins to stem the flow of blood. (Star Trek)
In 2372, Elim Garak suffered a nosebleed after being punched in the face by a Cardassian soldier during his unconscious state while experiencing the Great Link. (DS9: "Things Past")
Also in 2372, Jake Sisko suffered two nosebleeds while under the creative influence of Onaya. She offered him a handkerchief the second time. (DS9: "The Muse")