It doesn't currently work that way, and as Ann said, it might be a nice feature request. Make it part of the Project Manager perhaps.
Now, with some effort, you might be able to use OCR on a screen capture to get it into a usable format. I haven't tried it, but it might work.
The best solution might be to export it to a Final Cut Pro XML file and then just parse the XML file. I think a lot of what you want might be in there.