Was it coming out of one or both nostrils? I would check to see if there are any cuts or scabs in her nose. That could make it look like a bloody nose, this has happened to my mare before. If there aren’t any then there is clearly another cause. Since you said the cold cloth helped it could mean that it wasn’t just a cut. Are your workouts very hard? It isn’t uncommon for racehorses to get bloody noses after an extreme workout or race. This could mean EIPH (exercise induced pulmonary hemorrhage). If it was just your average workout though then she may have hit her head on something? If she was rolling and hit a rock maybe? I’m not sure... I would just keep an eye on it and if it happens again then I would call your vet.