Do horses get nose bleeds?

Yes - in fact they're fairly common, enough that there's a racing term "bleeder" for a horse that gets nose bleeds after a race.

Nose bleeds at rest are rare, and can be caused by foreign bodies or trauma to the nose, fungal infections, cancer, sinus disease and infections in the guttural pouch (again, often, fungal in nature).

Most commonly, as mentioned, the horse "bleeds" after strenuous exercise. This actually comes from the lungs - horses are prone to rupture of small blood vessels when they work particularly hard. (Abscesses and pneumonia can also cause this). Bleeders are generally treated with a drug called lasix - but it's banned in some countries as it also enhances performance and many trainers put all racehorses on it whether they bleed or not.