The 21-year-old Bristol, daughter of politician Sarah Palin, became pregnant at the age of 17 during her mother's campaign to become the US vice president.
She and Johnston were briefly engaged after news of the pregnancy, but have since split.
TMZ reports that Johnston is required to pay $1,750 ( £1,100) a month by court order, but now owes over $38,000 ( £23,923) in back child support.
According to Bristol's lawyer, Johnston has not made payments since June 2010.
The lawyer explained why they had been slow to enforce the order until now, saying: "We forbore pursuing child support collection in the (perhaps naive) hope that Mr. Johnston would do what men are supposed to do and pay his support obligation voluntarily.
"We were also concerned that garnishment or other aggressive collection efforts could exacerbate an already strained relationship."
Johnston confirmed on Wednesday (April 4) that he is expecting his second child with girlfriend Susan Oglesby. He has yet to comment on the child support allegations.
Bristol has signed on for a documentary series with Lifetime called Life's A Tripp, about the experiences of being a young mother.