No, it isn't possible. After all QuickEdit is for quick editing, essentially just short jobs, so you wouldn't normally need to Save and continue editing.
For more serious editing I suggest you use something like BBEdit or TextEdit (or various others). These have more features and are much better in the long run for more than a quick change. In fact since QuickEdit runs locally on your Mac the file still has to be downloaded and then uploaded at the end -- just the same as for an external editor, so there isn't much time saved by using it, especially if you have your external editor already open.
