Can a child break his forearm without any showing any swelling or discoloration?

Yes: The younger the child, the "softer" the bone and the greater the chance that there will be minimal swelling, discoloration, or deformity. Children's bones are more like fresh healthy branches that bend more than break, as a dried out twig might. Also, when the bones do deform they have a greater capacity to "remodel" whereby the body reduces the deformity as the bone heals.

Child forearm fx: It is unusual for a child to break a bone in the forearm without any swelling, but that being said it is not impossible. There may be a hairline fracture in a pudgy child which is hard to determine if there is swelling. Best physical finding is point tenderness. Sometimes xrays do not pick up fractures, especially with old equipment.