Just looking at my new 509 Sinisters here, if you use a thin, dull blade you can push the strap out the slot in the side swivels. Just start at one end and progressivly stuff the strap into the slot, gently pulling as you go. If that doesn't work, looks like if you snip the threads where the strap is sewn back on itself, you'll be able to undo the end of the strap (and not destroy it) to pull it out. Then you'd have to re-sew it back up if you wanted to use it again...
I have a quick strap on my 509 aviator. I didn't cut the old strap off, and the quick strap installed without any modification. Just pull the plastic pin out of the end of the strap. Its a good idea to keep the old strap as a backup if anything happens to the quick strap.