Yes if the mother has a cold and is in contact with the newborn baby then yes the newborn baby can catch the cold virus from the mother.
The cold virus that the newborn catches from the mother is usually unavoidable since the mother has to care for the newborn.
When a newborn gets a cold virus it will usually just go away on it's own within a week just like an adults or child's cold virus.
Sometimes you might need to take the newborn to the doctor for the cold but most times it goes away on it's own.