Understand that you want local WiFi control and NOT cloud based. There are WiFi and Ethernet controlled relay boards that can do just that. An example of one is at
https://ebay.to/2Uj7EsR for $20. You can access it via your router's WiFi using port forwarding and it has its own server to operate the relays which you can wire to a garage remote fob to control one or two doors. Most of the WiFi ones I've looked at require an app and the cloud.
My Stargate has an Ethernet connection set up as a static IP address with port forwarding so I can access it from anywhere and operate relays that are wired to the switch pads in a garage door opener fob. I've set up the relays to go on for 1 second to transmit from the fob to the doors (two relays, one for each door). The above device can do the same thing.
Because the Stargate receives and can act upon X10 commands I can also control it with Alexa using my Smartenit Harmony P2 device.
When I say "Alexa open double" the P2 sends N4-ON and the Stargate uses that as a trigger to operate the relay. When I say "Alexa close double" the P2 sends N4-OFF. To prevent inadvertently opening or closing the door I have a magnetic alarm contact on each door connected to digital inputs on the Stargate so the open/close status is known and used as a conditional so if N4-OFF is received and the door is already closed the relay will not be triggered. The Stargate will also close either door if left open for more than 15 minutes after the alarm is set and text me whenever the doors are opened. I'm sure you can interface to HG to do the same.
The doors are named 'Double' and 'Single' and sometimes Alexa hears 'Bible' when I say 'Double' which results in a random passage being recited rather than the door operating.
My Doors are Liftmasters with the MyQ hub. The MyQ hub is compatible with Google and IFTTT but not Alexa. They forced Amazon to delete the unofficial MyQ skill that worked great but wasn't theirs. The Google and IFTTT compatibility are "free initially" which indicates that they'll start charging at some point. The app works fine but is cloud based.