Only if you have the appropriate level of privacy settings enabled (and extensions installed) in your browser. Your IP address actually has very little to do with ID-ing you, since most trackers will use hundreds of different fingerprinting methods to create "shadow accounts" of you using things like your system information, screen resolution, installed locales, etc.

This doesn't mean a VPN doesn't help, though. Just pointing-out that you probably won't be asked if you're a bot if you go on Google while logged-in to a Google account, regardless of whether your VPN is on or not.