If your child has just turned 1 year old then no the over the counter cough medicine cannot be given to the 1 year old.
Once the child reaches around 14 to 15 months of age you can usually give the toddler some honey dissolved in some water to help with their cough.
Or speak with the toddlers doctor about some medication that they may know about and can get for you.
Cough medicine that comes over the counter should not be given to any children under the age of 4 so if your child is 3 years old or younger the cough medicine is not safe for them.
Also for safety reasons and to avoid choking you should only give cough drops to a child that is 5 years old or older.