Q: What's the Intent to open DND settings? I'm sure I'm overlooking something in the Settings class documentation. What Intent can open the Settings app in the "Do not disturb" section?
I expected it to be the ACTION_NOTIFICATION_POLICY_ACCESS_SETTINGS action, but that is only for the screen which lists which apps have requested DND access.

A: Update
Looking at the AndroidManifest.xml for the Settings app there is an Activity Settings$ZenModeSettingsActivity already from Android 5.0.
To send the user to the "Do not disturb" screen you can use the action android.settings.ZEN_MODE_SETTINGS like this:
try {
    startActivity(new Intent("android.settings.ZEN_MODE_SETTINGS"));
} catch (ActivityNotFoundException e) {
    // TODO: Handle activity not found
}

Original answer
It looks like there are no screens in the Settings app (at least on Android 6+7) where you can enable/disable DND. It seems like this is only available through the settings tile (and can be disabled in the dialog when changing the volume).
I have a Samsung S6 (Android 6.0.1) which has this screen, but this is probably some custom Samsung changes. The screen is represented by the class com.android.settings.Settings$ZenModeDNDSettingsActivity which can be started by any app. This might be of help for some people out there.
AndroidManifest.xml for Settings app for Android 6+7:

*

*https://android.googlesource.com/platform/packages/apps/Settings/+/android-6.0.1_r68/AndroidManifest.xml

*https://android.googlesource.com/platform/packages/apps/Settings/+/android-7.0.0_r6/AndroidManifest.xml

A: You have to use the following Intent: ACTION_VOICE_CONTROL_DO_NOT_DISTURB_MODE and then pass a boolean through EXTRA_DO_NOT_DISTURB_MODE_ENABLED. 
Make note that the documentation specifies the following: This intent MUST be started using startVoiceActivity.
