All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Salmaan Al-Faarisi was nicknamed Abu 'Abdullaah. He did marry, as Ibn Hajar stated, "He married a woman from the tribe of Kindah." However, we have not come across any statement that he had a son named 'Abdullaah in any of the books of biography and history.
Nonetheless, the biographers who wrote about his life stated that he had three daughters, and there is no religious obstacle for a man or a woman to be nicknamed even if they had no children at all, or even if they did not have a child with the nickname they were given. 'Aa'ishah said, “O Messenger of Allaah, all my female friends have a nickname.” He, sallallaahu ‘alayhi wa sallam, said: “Take the nickname of your son 'Abdullaah (meaning the son of her sister).” [Abu Daawood] Furthermore, Anas narrated, “The Prophet, sallallaahu ‘alayhi wa sallam, used to socialize with us (when we were children) and he, sallallaahu ‘alayhi wa sallam, used to call my little brother: “O Abu 'Umayr, what happened to the small bird, (giving him a nickname)?” [Al-Bukhari and Muslim]
Allaah Knows best.