Should hydrogen peroxide be applied on a bloody nose?
No, I don't think oxygenated water does anything to stop bloody noses. But holding a wet cotton swab with pressure to the inside of the nose might.

The vast majority of bloody noses in healthy individuals arise from one specific area in the nose (on either side):



In the mucosa of the nasal septum (the cartilagenous structure separating the nose into two sides), there is an area where several arteries "meet", giving it an exceptionally rich vascular supply, called Kiesselbach's plexus. It is on the anterior surface, and exposed to dry air and trauma (even such as might be sustained by a bad cold). Bleeding typically occurs when the mucosa erodes for any reason, and the capillaries and venules (and sometimes arterioles) become exposed and subsequently break. The result is the familiar bloody dripping of a nosebleed.

The most common treatment is direct pressure (squeezing the sides of the nose together) for 5-10 minutes. This works because putting direct pressure on any bleed stops the blood flow long enough for a clot to form and the arteriole to close down.
If needed, more aggressive control can be achieved by a professional.

If you happen to apply pressure with the wet cotton swab to a large enough part of the plexus (or somewhere near the spot from which your nose is bleeding), your nosebleed will stop. But it really doesn't matter what you use to wet the cotton swab (plain water, oxygenated water, carbonated water (ouch!), or other. (Noted: OP added that the liquid is hydrogen peroxide. This is not a good idea as it is an irritant and actually can cause mucosal damage.) Sufficient pressure and clot formation (and subsequent tenderness with the nose) is all that is needed.*

What doesn't work: 


icepacks to the neck
leaning your head forward (increases blood flow) or back (may cause the blood to flow into your throat)
lying down (increases blood flow)
frequent 'peeking' to see if bleeding has stopped
a dry object placed against the area of bleeding (it will break up the clot when pulled away from the septum)


Aftercare and prevention is another subject.

Note: The safest way to end a common nosebleed is by squeezing as much of the 'rubbery' portion of the nose together for up to 30 minutes if necessary. Putting something into your nose poses the risk of infection.

*more serious or recurrent bleeding needs medical attention