You should be able to bring the Quicktime file into Final Cut (and probably other programs as well), add the watermark as an overlay, and then output the composite to a new Quicktime file. Just be aware that if you're bringing in a degraded Quicktime file (meaning a compressed one for Web use, rather than a full-resolution one,) you will probably have a little bit of further degradation by rendering it out again.

If you have it, your best bet would be to use the original project file from which the edited video was output, and just add the watermark as an overlay to the edited project and output that as a new Quicktime file.