Leave the baby in until he's all dry and fluffy, then put him in the brooder with a small stuffed animal. If you don't have an animal, you can take an old washcloth, roll it up, fold it in half and rubber-band it to make a little donut. We had one hatch 48 hours before any of the others, and he just loved sleeping under the lamp in his little donut. Once the others came, he was happy. But, Buttons are tiny, and will dehydrate rapidly. They need to have food and water no later than 24 hours, or they will weaken. The sooner the better. They're not like chickens where they take 3 days to absorb the yolk... quail will eat and drink on their own right away.
Congratulations on your baby... I do hope another one hatches, though, so he won't be lonely. Do you have more eggs coming?