No, Emperor Nerva did not have a biological son (or any children at all) when he adopted Trajan in 97 AD. In fact, there is no historical record indicating that Nerva ever married. [1]

His lack of a natural heir—combined with his lack of military backing and an aggressive mutiny by the Praetorian Guard—forced him to look outside his bloodline to secure his position and prevent a civil war. He chose the highly popular general Trajan as his adoptive son specifically to appease the Roman legions and ensure a stable succession. [1, 2, 3, 4, 5]

If you are interested in this period of Roman history, I can share more details about:

• The mutiny that forced Nerva's hand
• How the Adoptive Dynasty (the "Five Good Emperors") operated after Nerva
• Trajan's immediate actions upon taking the throne [1, 2, 3, 4, 5]