"Is it true that supplication for a living Muslim is made by mentioning the father's name, and for the deceased by mentioning the mother's name? "

What is established in the Book and Sunnah is that a person is called by their father's name, not their mother's, whether in life or after death. This is mentioned in the Noble Qur'an and in authentic Hadiths from the Prophet, peace and blessings be upon him. If a person does not have a known lineage, they are called after their death by the name they were called during their life. The Hadith that suggests one should be called by the mother's name or attributing someone who does not know their mother to Eve is considered weak.