Well, yes and no
First of all, the subdomain can redirect to google.com and back, loading it in a first-party context, and then redirect to your main domain page. I guess Google here would be a “second party”. https://learn.microsoft.com/en-us/azure/active-directory/dev...
Second way is that subdomain can load an iframe from google.com and the iframe will send a postMessage to the enclosing page, which will send a request to the server to set a cookie.
As long as you logged in ONCE in google (let’s say in a popup, or maybe using ITP’s click-to-login) then it will store the session cookie this way. And after that it can meep a cookie around for 10 years and track this guy across all the subdomains where he signed in once.