In Scar Tissue, Anthony mentions that he asked her to watch RHCP’s performance on the TV show, Saturday Night Live where they played ‘Under the Bridge’ and ‘Stone Cold Bush’. That performance became infamous because it’s the one where Peppers’ guitarist, John Frusciante, played ‘Under The Bridge’ differently thereby upsetting Kiedis who kicked Frusciante and swung a mic at his head as a result!
Apparently Kiedis was even more upset because his new girlfriend, Coppola, was watching. That SNL performance was in February 1992.
In Vogue (June 1994), there’s a reference to Anthony Kiedis and Sofia Coppola dating and it gives the date as 1993. The couple also appeared in a short film for Details magazine; the release date of that is again 1993 (although the film could have been released later it does seem to confirm the 92-3 date).