I tried to add my Google calendar to Kontact, but when I try to configure the account, I am allowed only to type my username. After that, Google informs me,

Couldn’t sign you in
This browser or app may not be secure.
Try using a different browser. If you’re already using a supported browser, you can refresh your screen and try again to sign in.

Is there a way to make this work?

Once upon a time I apparently authorized Akonadi Resources for Google Services to use my Google account, so if there's some way to make Kontact recognize that, I'd be delighted. However, I don't seem to find that resource anymore, not even in Akonadi Console.

Same problem for me. I've choosed Chromium as default browser, should be the best to access a Google resource. Possible ways to investigate:
(a) Login to your Google account, look if there is a related setting
(b) Something in Chromium configuration. may be security settings
What should we expect? May be that Google initiates a menu, asking: "Do you want to allow Akonadi to access your Gooogle contacts"

Thanks for the reply. Sorry for my own late reply, but I may have figured out my problem (but not yours). I'm using Fedora, which is still stuck at libkgapi 20.08.3. The latest version is 20.12.1. I'm not 100% sure that that's the problem, but this bug in libkgapi looks similar, and was solved some time ago, so I suspect my problem is that Fedora needs to catch up.

I came across this issue yesterday. Thought I'd see what would happen if I copied the URL from the browser's bar and paste it into Firefox. It actually worked and I could access my Google account.

Faced the same issue. Although simple solution that worked for me is was, I opened the url in firefox and it worked. I was able to login to the account without any issue. Hope it helps.