Frankie Green is the son of Kara Green (née Foster) and Danny Green. He was born off screen sometime between the Season 2 finale and Season 3 premiere. It is implied that he was born earlier than expected, which caused Danny (stationed in Asia at the time) to miss his birth.
Frankie is first seen in the Season 3 premiere with Kara, Debbie, and Valerie as they work in Kara's new office in the St. Louis White House.
Danny finally holds his son for the first time in the Season 3 finale.
During Kara's pregnancy[]
Kara first finds out about her pregnancy while she underwent treatment as a test subject for the cure. She suffered from a seizure after being injected with the Red Flu, which was originally thought to be an effect of the trials. After stabilizing her, Dr. Scott and Quincy Tophet realize that the seizure couldn't have been a result of their testings. After running another blood panel, they find out that Kara is pregnant. Chandler breaks the news to Danny, so he could visit her and keep her company as her health declined. Once Dr. Scott found the solution to why her cure wasn't working, she injected the test subjects with a new batch, and Kara and the rest of the volunteers became immune. Dr. Scott later assured the couple that not only was their son alright, but he would be born immune to the virus. (“Trials”)
While the Nathan James was taken hostage by Amy Granderson and the Baltimore police, Dr. Hamada had Kara taken to the Baltimore facility so he could try and extract the cure from her baby's stem cells. She was strapped down into a medical chair, but before Hamada could begin, a medical assistant loosened Kara's hand strap so she could grab a syringe and stab Hamada in the heart. Moments later, Nathan James crew members bust in and help Kara out. (“Unreal City”) (“Fight the Ship”)
Kara is around 16 weeks pregnant before the time skip between Season 2 and Season 3, so her pregnancy isn't visible through her uniform.