The easiest and cheapest method is to blow the car's horn and wait a few seconds before opening the garage door.
The next easiest and cheapest option would be to use a wireless door bell and put the button in your car. It will only cost $10. But, it will require the driver to press the door bell first, wait for the desired time, and then press the garage door opener. Lots of opportunity for human error and not automated.
The over-engineered home automation method could utilize a LiftMaster Universal Garage Door receiver to receive the car's garage door opener button press.
But, instead of feeding the receiver's output to the garage door, you would feed it to a smart sensor, like a Z-Wave Door/window sensor to notify the home automation controller of the event.
The home automation controller can then run a scene that sounds the alarm in the garage, waits for a period of time, and then sends a garage door open command on a different channel, like a Z-Wave dry contact relay, or a ratgdo.
For the purpose of "emergency" access, in case the home automation system has failed, I'd also associate the second or third button on the garage door remote directly to the garage door. You won't have the alarm but you'll be able to open the door.