^There was always the first two notes of the Klingon theme" theme/motif that popped up in the Klingon-centric episodes of TNG... but I disagree with you about not using the DS9 theme. At the time it would have been wholly appropriate to do so. Worf was and had been a regular character on DS9 for over a year by the time First Contact premiered; further, the Defiant itself was a mainstay from that show as well.
Likely, Jerry Goldsmith didn't do it (if it had even occurred to him or been suggested) because the production would still have had to pay a nominal fee (and then royalties) to Dennis McCarthy for using his theme for DS9 in the film. Goldsmith lifting his own Klingon theme worked fine though.