Sorry, but your breeder told you nonsense. A bird's nail is structurally no different than a dog's or cat's nail (or a human fingernail, for that matter). Cutting the tip of the nail (done properly, with a properly designed sharp tool) is no more harmful to your bird than it is to your dog or cat. And while wild animals don't need nail care, in captivity they sometimes do. They just can't always be active enough to wear their nails down properly. But if you don't know how to trim a nail properly, you could cause some damage - or worse, cut into the quick and trigger bleeding.
Beaks are another matter entirely; most birds do chew enough to keep their beak properly worn down, and beak trimming is something only an expert should attempt. Since you're concerned about your conure's beak as well as his/her toenails, why not make an appointment with an avian vet who can check the beak and make sure everything's OK, and also demonstrate to you the safe and proper way to trim your bird's nails in the future?