I have the Wired 2nd Gen Doorbell, and it's the only doorbell in my house. After enabling Quiet Time and confirming that it shows enabled with a countdown timer on the screen, the indoor mechanical chime still rings. The only way to disable the mechanical chime is to disable "Ring indoor chime" in Settings > Doorbell.
Also, as many users have already suggested, it would be nice to have a way to schedule or use Google Assistant to either enable Quiet Time or toggle the "Ring indoor chime" setting. For those with kids and/or dogs, it would be a lifesaver!
Sorry to hear you're experiencing this, as I can imagine it has been an inconvenience. I would love to look into this further for you. When you turn on Quiet Time, are you also checking to make sure that Indoor Chime option is turned off? If it is on, turn it off and try again. I would also restart your device if you haven't already. You can learn how to restart it here, or factory reset it via the same article.
I was definitely asking you if you were seeing the setting turned on when Do Not Disturb is on, to see if it is malfunctioning or not. Are you seeing it showing as on when you have Do Not Disturb on? Or is the option to turn it on and off greyed out when Do Not Disturb is enabled? Please let me know.
@Brad I'm more confused now. Are you referring to the Do Not Disturb system setting on my phone? That shouldn't have anything to do with this issue. But to entertain your suggestion, the Doorbell settings are available (not grayed out) regardless of my phone's Do Not Disturb setting.
The issue boils down to software (Google Home) that is not correctly controlling hardware (Nest Doorbell). Here are the exact steps to reproduce the issue:
As I mentioned before, the only way I've found to disable the indoor chime is the "Ring indoor chime" setting, which is accessed with the following steps. However, this is an indefinitely long setting and you have to remember to come back to the app to reenable the indoor chime. The Quiet Time feature would be way more convenient if it works (and if it could be automated using Google Assistant or a schedule).