Dumbledore meets her at the Hog’s Head about taking the position, although he is disinclined to continue offering Divination at all. Trelawney then makes a prophecy about a baby who will be born at the end of July, a baby who will be able to defeat the Dark Lord – and the prophecy is overheard by Severus Snape, lurking in the hallway. Though Dumbledore claims Snape heard only the first half of the prophecy (OP37), Trelawney later reports that he was there at the end of it as well, although he had been interrupted in his eavesdropping by Hog’s Head owner Aberforth Dumbledore, who was about to throw him from the building (HBP25). Trelawney thought Snape was there merely to get job interview tips, so the fact is she hardly understood anything that was going on.
This interview had to have taken place sometime after July of 1979 in order for it to apply to Harry's birth in 1980, and Dumbledore describes it as a "cold, wet night" (OP37). So it was likely made in the winter or early spring of 1979-1980.
We also know that Trelawney, in early September of 1995, claimed she had held her post for "nearly sixteen years" (OP15), which would imply that she was hired in the middle of a school year. And then, Dumbledore says in June of 1996 that the interview was "sixteen years ago" (OP37). So unless Dumbledore is wrong, the interview must have been in the first few months of 1980 - sixteen years before Dumbledore's statement, a slightly exaggerated 'nearly sixteen years' before Trelawney's, and during the time of year most likely to have a "cold, wet night."