In the movie Men of Honor (2000), Carl & Billy are challenged to hold their breath in an underwater helmet.
After 4 minutes, Billy started bleeding from his nose.
Does blood really come out of your nose if you cannot breathe?
I doubt something like that scene would happen, but I do think it's possible. Nosebleeds can occur from high blood pressure, which I think is what would be happening there.
However, nosebleeds do effect divers (typical new divers, but still), from here:
New divers frequently report nosebleeds after diving primarily because they are unaware of the importance of equalizing the sinuses and middle ears (via the Valsalva method). The barotrauma that is produced when the sinuses are not cleared can cause blood vessels in the lining of the nose to burst. These vessels lie very close to the top of the mucous membrane lining in the nose and sinuses, and the blood can come from the lining in either. This type of barotrauma, generally the result of air being trapped within the sinuses, is not always painful, though the presence of blood can be disconcerting to a new diver. With this type of injury, blood can run down the back of the throat or pool in the sinus below the eye and emerge at a different time. It can also act as a growth medium for bacteria and result in sinus infections.
That said, I think it might be related to what's found here:
Nosebleeds are relatively common, especially in children, according to NHS Choices. They occur when the tiny blood vessels in your nose are disturbed or become inflamed due to infection or increased pressure -- such as when you blow your nose or even hold your breath when swimming.
In short, it seems entirely possible and there are many ways that Billy's nose could have started bleeding.