I'm not sure of any health-type issues of keeping an adult bearded dragon in a 20 gallon, although I can say with a fair amount of confidence it'd be like keeping a dog in a bathroom it's entire life. When the beardie becomes fully grown, it won't be able to turn without hitting one side of the tank or the other, or even both.
You can do it, sure, but it's not recommended. If you could ever get enough room in your room for at least a 40 gallon, that would be best.
Also, take precaution when you take your beardie outside!
Don't let him/her eat any strange insects or veggies and keep an eye on the sky, as a hawk or eagle would see the beardie as a potential snack. There are lots of stories of someone turning their head for a minute and when they look back, the beardie is gone. So while it's always a good thing for fresh sun, just keep a close eye on him/her.