Nayoung is the ex-wife of Seok-woo & mom of Su-an who lives in Busan.
Prior to Train to Busan, Nayoung & Seok-woo got hitched & had a baby girl named Su-an, but she left Seok-woo after because he was starting to become obsessed with his work & becoming selfish.
She was first heard talking to Seok-woo about their divorce and Su-an. Nayoung also tells Seok-woo that Su-an wants to come to Busan by herself on her birthday & reminding him that it's the next day.
Then, she's kind of heard talking to her daughter when Su-an is in her bedroom. We can only assume Nayoung is convincing Su-an to let her father take her to Busan. Su-an then hangs up when Seok-woo shows up.
After that, Su-an and her dad are driving to the station and Su-an tells her that Seok-woo is driving her and that she'll see her in Busan very soon.
It is unknown what happened after that, as her phone was off when both Seok-woo and Su-an try to call her. We can only assume that she is probably long gone or she is busy. It is also unlikely that Su-an ever reunited with her.
Relatives: Seok-woo (Ex-husband), Su-an (Daughter) Mrs.Woo (mother in law)