Dogs do have nerves in their claws and nails.
Dogs have a sensitive area inside their nails that is called and known as the "quick".
The "quick" in a dogs nails contain nerves and blood vessels.
And cutting into the dogs "quick" when cutting the dogs nails can cause bleeding and pain.
So it's important to trim the dogs nails very carefully to avoid cutting the "quick".
And if you're unsure of how to cut the dogs nails correctly to avoid the "quick" or you're nervous you may cut into their "quick" and cause bleeding and pain you should have a vet or a dog groomer or someone else that knows how to cut the dogs nails properly instead.
I'm pretty confident myself in cutting nails and could do it, but I prefer to pay a dog groomer to cut my dogs nails to be safe.
If you do cut a dog's "quick" it will bleed and cause pain as the "quick" contains blood vessels and nerves.
To stop any bleeding if you accidentally cut the dog's "quick", apply some cornstarch, flour or styptic powder directly to the dog's nail with pressure.
And keep the area clean and monitor it for any signs of infection, such as redness or swelling and if the bleeding persists or an infection is occurring, you should contact the vet.