Assuming North America…
There are 12 volt vehicle power USB adapters available, which plugs into the available vehicle socket variously called either a power port or a cigarette lighter.
I’ve not encountered an Apple Watch charger puck that wasn’t connected to either USB-A or more recently with USB-C, and from there to a detachable 110V wall-socket charger, so I’m not sure what charger puck and cable you have there.
Apple sells both (older) USB-A and (newer) USB-C pucks, and you’ll want whichever one matches the vehicle car charger you’ve acquired.
Apple Watch can be charged with its puck and cable powered from either USB-A socket or USB-C; whichever matches.
Various vehicles can also have built-in USB-A or USB-C sockets, and some vehicles with 110V wall sockets, so check your owner’s manual for that.
If not in North America, other areas work similarly, but sockets used and the voltages supplied can vary. USB-A and USB-C are the same world-wide.