Not something that can be done in/with Connect‑IQ, unless you want to run a dedicated watch app all the time that gives you access to such a ‘button’.
The idea of an SOS button, such as that which may be found on my dad's simple mobile phone handset designed for the elderly, cannot just be translated to Garmin consumer fitness wearables, because:
You do not (and will not) get a dedicated hardware button, on any existing Garmin device models, that will allow the user to manually trigger the call-for-help procedure irrespective of which mode the device is in or what it is doing.
Garmin devices, on their own, have no hardware-enabled capability to connect to and communicate over a cellular mobile network, but must rely on a phone handset to which a Garmin device is connected (via Bluetooth Low Energy mode).
Even though some Garmin devices can connect to Wi-Fi networks (after access details have been configured via the Garmin Express application on a computer), it would be silly to assume that a Garmin device will be connected to a known Wi-Fi network (pre-authorised for access) in an emergency in which the call-for-help procedure is useful for the user to manually trigger.
Now, more specifically in response to sergeii13's detailed proposal,
To assign a function to a hardware button, the watch needs to be running a CIQ widget or watch app designed to do that, and not tracking a timed activity using a built-in activity profile (or app) such as Run or Bike. That makes LiveTrack irrelevant. You cannot leverage LiveTrack as the communications platform for calling for help, or use the list of people you have invited to view a particular timed activity on LiveTrack for any other purpose.
If the user has to fiddle with the watch (to stop and/or discard a timed activity) to activate the widget or watch app, in order to gain access to the call-for-help procedure eventually by a button-press, then the user would be better off just activating an app on his/her smartphone handset that calls for help. I cannot think of a Bluetooth Smart-enabled handset (capable of running Garmin Connect Mobile and connecting to the Garmin device) that does not also have built-in GPS capability. Yes, it's conceivable that after a bad fall or crash, the user is effectively immobilised but can still reach his/her wrist, even though he/she cannot pull out and operate the smartphone handset. However, the handset will still need to be functioning sufficiently, connected to both the cellular mobile network and by Bluetooth to the Garmin device within range, have adequate battery power, and is running the relevant companion app. All those pre-conditions put together are more likely a bigger ask than being able to retrieve the (sufficiently functioning) handset and start an app that operates independently of the Garmin device.
