My reaction was exactly the same as Martin's. I suspect the sentence was "It was the epicenter of the San Francisco arts movement." No apostrophe or 's' is needed, because "the San Francisco arts movement" is a compound noun. It's easy to miss little unstressed words like 'the' in connected speech. The vowel was probably so reduced as to be virtually inaudible, and you didn't hear the 'th' sound because it seemed to merge with the 's' of 'San'. Listen again and see there's a hint of a 'the' before 'San'.

I agree with Martin and Su.ki. I will add that it also could have been "the epicenter of San Francisco's arts movement" and then the apostrophe-s would be used after the city name.

I am guessing the sentence was "It was the epicenter of the San Francisco arts movement." Maybe you did not hear the "the"