[joosters]

No. iCloud backups don't contain keychains, where the Google authenticator stores its seeds.

[chatmasta]

Doesn't iOS keep keychain synced with iCloud keychain? Or at least, it's user configurable. I'm pretty sure I opted out of it.

Good news is according to apple [0], you can protect your icloud keychain with a six digit code required to move the keychain to a new device.

[0] https://support.apple.com/en-us/HT204085