It can't have been later than 2347 (Setlik 3 massacre and beginning of Cardassian-Federation War) but is there any other reference to when the two governments made official first contact?
There's no main canon description of the Federation's first formal contact with the Cardassian Union. They do encounter Cardassian individuals in Star Trek: Enterprise — a Cardassian Borg, and what appears to be a Cardassian body at the automated repair station in "Dead Stop" — but these obviously don't answer the question of first official contact.
There are some (conflicting) accounts in various Trek novels and comics.
Per Memory Beta:
By some accounts, first contact between the Federation and the Cardassian Union occurred circa 2321, on the human colony world of Juhraya. (VOY - The Brave and the Bold, Book Two novella: The Third Artifact)
This conflicts with Day of the Vipers, which states that the Cardassians were already engaged in "pointless, unresolved conflicts" with the Federation by 2318. This implies a first contact date sometime before or during 2318.
The first official talks between the Cardassians and the Federation took place in 2327, however the Cardassians were using the talks as a pretense to disrupt relations between the Federation and Legarans to make Legara IV susceptible to annexation by the Cardassians. Fortunately the plan was uncovered and the Federation placed extra Starfleet protection around Legara. (TOS comic: "Enter the Wolves")