When you encode to H.264; you need to set the audio to AAC.
It defaults out of FCP to Linear PCM.
This is a common mistake, and easy to fix.
BTW - you technically don't even have to change the file to .flv; but perhaps your client will feel better if you do. Flash players easily read h.264(.mov).
Thanks all. I will change the audio and see how that works. I still find it odd that the original file plays fine then when you change the extention it won't play the audio in the Flash Player. Yet it I change the extention back it plays fine in Quicktime. Doesn't make sense to me.