The post, which was made on Friday, has been shared over 13,000 times and has attracted international media attention.
In it, Kerryn Leigh wrote: "we want nothing from him but possible conversation. Who knows, I just need to do this for her."
Kerryn Leigh said after giving birth to her daughter she had given up on finding Rodney.
"I was (almost) 40 years old with a baby in my arms with a man I didn't know," she told Fairfax media. "I was coping with what I had. I would have let him know. He deserved to know."
She described Rodney as a surfer living on the Gold Coast with long scruffy hair, blue eyes and is believed to be in his late 40s now.
Since creating the post on Facebook, Kerryn Leigh has been contacted by several women with possible leads but no men had come forward, Fairfax reported.