The 19-year-old and former girlfriend Bristol Palin became parents to son Tripp last December, but they split shortly after his birth.
Speaking to The Insider, Johnston said: "At the end of the month I'm going to file for joint custody of Tripp."
He reportedly alleged that his decision to file for joint custody was sparked after he had difficulty getting access to the 10-month-old child.
"It's not working. I'm done. It's going to have to go to court," he added. "They just finally pushed me over the edge."
A representative for Palin has yet to comment, although People magazine reports that her attorney issued a statement to The Guardian in October on the issue of custody.
Lawyer Thomas Van Flein said: "Other than noting that Tripp's father is always welcome to visit his son, we are unable to respond to these allegations as it is inappropriate to discuss child custody matters publicly."
> Palin "appalled" by Johnston remarks