Nose picking. Toddlers pick their noses whether they confess to it or not. Further, they have a hard time understanding how to clean/blow the nose. You don't need to be overly agressive.. Try these simple things: saline nasal spray 2-3 times a day to keep the nose moist; humidifier at night; a vaseline or neosporin ointment applied gently to nostrils before bed at night.
It can happen easily. Nosebleeds can happen at any age. Contributing factors include dryness, irritation (including putting fingers in nose, which toddlers do!), allergies, and injury. Ask your child's doctor if it's ok to use a humidifier in the bedroom as well as putting saline gel in both nostrils 2-3 times daily. If it's no better with that, your child needs to be seen.