Turner and baby mama Maria Grissom are battling in court over Turner's daycare payments for their daughter. According to documents filed in Georgia, Grissom claims the Atlanta Falcons superstar NEVER pays his daughter's school money on time is demanding he pay her up to four years in advance ... to the tune of $59,121.

But Turner is firing back ... claiming he DOES pay the school on time and it's Grissom who is in the wrong ... blasting his personal life drama all over her Twitter account in violation of their court agreement.

In the docs, Grissom admits to the Twitter rampage, but is adamant that it's Turner's fault ... claiming she had to defend her reputation ever since one of Turner's OTHER baby mamas (he has four in all) was involved in a "ghetto fight" at his house that made national news.

Turner is requesting to pay all the money he owes directly to the day care and NOT to his baby mama, and for her to stop talking trash about him on the internet.