However, there are occasions when the beak gets damaged, immediate first aid is needed and a vet isn't readily available. Damage to the beak is rare but does occur. Often it is limited to the tip snapping off for one reason or another. It's happened to both of my birds more than once over the years.
Like the toes, the beak is a living thing (except for about 1/3 of it at the end) with the usual nerve-ending feedback that prevents damage-causing pressures when biting (just as it would be difficult for you to break your jaw by biting down too hard - possible but very unlikely.) As the tip grows or wears down, these nerves gradually extend or retract to keep the same quarter inch. But if you clip more than a quarter inch or it breaks, the nerves are exposed - ouch! But they naturally die back and retract over a couple of days and all is well.
HOWEVER, at one point my Greenwing snapped at a toy I had, hooked it, and when I reflexively twisted away, chipped off the 1/4 inch tip at the end of his beak. Panic ensued (on our part - he was just annoyed) - blood dripping off the end of the beak, him shaking his head and throwing what looked like gallons of blood all over. Scared.us.to.death - but corn starch stopped it almost imemdiately although we had to towel him to apply it.
Of course, this happened on a Friday night and we couldn't get him to a vet until Monday. He was a very sore bird that weekend, eating mush out the side of his beak and flinching when the tip touched anything or when he dipped it into cold water. By Monday it was noticeably less sensitive, but I was worried about it because it had an inverted V notch at the end. The vet ground it off saying it was just possible (not likely - but possible) that once the sensitivity went away (as the pulp/nerves pull back), if he bit down on something at the tip, it could continue to split like a wedge in a piece of wood. So he rounded it off and a week later all was back to normal.
I've since noticed that the beak doesn't 'gradually' wear down so much as little pieces on the end continually chip away. Sometimes it's rounded and smooth, sometimes uneven with a chip off one side or the other (around 1/16 inch). Once, the B&G snagged a bath towel, I heard a 'pop' and a 1/8th piece of the tip broke off and peeled back along the top, but stayed attached. He'd shake his head and it would flap around. I think he could see it too - I swear he was cross-eyed at one point. He finally let me pull it off. No blood and no sensitivity.
None of the above is meant as a warning or to scare you. Just experiences we had. If a minor accident happens, give him soft warm food for a couple of days until the sensitivity goes away. I never trim our B&G or GW beak - only their toenails. My B&G right now has an uneven tip (as opposed to split), but we'll just watch it for a week or so to make sure it evens out. Any sensitivity though and off to the vet.
If, however, more than a 1/4 inch breaks off or if you can'g stop the bleeding or if the side of the beak gets punctured (possibly by another bird) - these are not minor. See a vet as soon as possible - preferably one that is avian certified who will know something about beaks.