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DearCustomer- The support should stop automatically but I always go by the theory that people do what you inspect rather than what you expect. I always think it is best to file a motion to terminate support on the date it is supposed to stop. That way there will be a court order sent to the child support offices.
If you rely on the child support offices to stop the payments and they don't then you will still have to file a motion and then try to collect the over payments. That is not always a pleasant experience trying to get some government office to give you money.
You will have to create your own motion using the heading on your case with the court name at the top etc. You can just say "now comes (your name) and respectfully XXXXX XXXXX order by this court to terminate child support in this matter as of (your daughter's birthday). Pursuant to the child support order in place at this time support is due to terminate on her 19th birthday".
Sign your name, file it with the court and send a copy to the mother. There is nothing else for her to "come after" you for if that is when the support is to end.