It happened around 6pm Saturday in the 2200 block of Post Office at 22nd.
According to Galveston police, a woman was walking with a child in a stroller when stucco fell about 20 feet from beneath a balcony and hit them.
Both were taken to the University of Texas Medical Branch. Investigators said the child's injuries were not life-threatening, but the woman's injuries were more severe.
Police did not release their names or the age and gender of the child. We will bring you updates on their conditions as soon as we get them.
Officials believe the bottom of the balcony may have had water damage, causing the pieces to fall.
City officials shut down part of the sidewalk after the accident. It will remain closed until the building is inspected.