No, it won't become part of your Google Wifi mesh, this would create a new Wifi network, separate from your Google Wifi setup. Even if you use the same network name as your Google Wifi, it is still a separate Wifi network which could cause performance problems.
I'd instead recommend getting an outdoor rated Wifi extender/repeater. You want to look for one that has a "repeater" mode and not "AP/router" mode. Repeater mode amplifies and extends your existing Google Wifi network with the same Wifi network name. Ones that only support AP/router mode, still do that but by creating their own separate Wifi network.
Use Nest Wifi or Google Wifi with wireless extenders - Google Nest Help
