Casablanca, of course.
I had not read any of these other comments but as I was just watching this episode now I noticed the final scene with Quark and Garak (Andrew Robinson)... A very Humphrey Bogart/Claude Rains moment. The discussion they are having is very similar.
Of course what makes the episode is the involvement of Mary Crosby... they did quite a remarkable job with her Cardassian makeup.
And the other guest stars Michael Reilly Burke, Heidi Swedberg, and (RIP) Edward Wiley (Who was also Klingon Governor Vagh in the TNG episode The Minds Eye).
If only we had somebody to represent the Sydney Greenstreet and Peter Lorre characters...
But there is of course a "big silver airplane" waiting on the runway for Hogue, Rokellen and Natima.
I'm surprised I have never noticed this all these years until just now.
I appreciate that somebody else did notice it, the homage is strong and appropriate. And it reminds us that Star Trek has its ties to classic Hollywood going all the way back to the original series being filmed in Desi and Lucy's lot...