You will never notified of any system update in your browser. It will popup a message on top of whatever you're doing & show a notification in the notification bar.
Also, you don't necessarily have to, but I'd recommend clearing your browsing data. At least the Cache and History portions to prevent any other security concerns.
The previous time this happened, I uninstalled Chrome and then reinstalled it to no avail. It doesn't happen every day but it's extremely annoying. It just randomly launches a website when I'm in the middle of doing something else on my phone.