Nope, not for Chrome. It has it's own, "sandboxed" player. That's why I started using it. Unfortunately, there are/were other security problems with it, so I'm back to Safari. I still use Glimmerblocker that removes most "animated" (Flash) adds in all browsers.
I suspect you still have the Adobe Auto Downloader stuff still installed somewhere. I'm not sure where it is but it will probably continue to alert you of updates to Flash and other Adobe stuff (Reader?). Strangely, I have an "Adobe Flash Player Install Manager.app" when run it shows:

App name in Finder/PathFinder:

Typical Adobe "logic"? 
I think the functions that 'beg' for your attention are in the "/Library/Adobe/AAMUpdater" or "../Adobe/AAMUpdaterInventory". There are LaunchAgents as well as LaunchDaemons that do things behind your back!