That's your gas adapter, so you'll leave the part that says Tweco in the pictures installed on the machine. Separate the connector just downstream of the gas valve, and that'll be the fitting you are looking to match. The welding shop will be able to get you adapters for just about anything, they're inexpensive brass fittings, but the one you have should be easy enough to find that adapting shouldn't be a problem. I upgraded my Millermatic 35 to a 251 about 15 years ago, so I don't remember the finer details about the machine, but I think your setup is about the easiest and lowest cost machine out there to find good used parts for. I'd personally look for used equipment, and just replace the liner, tips, and nozzle if needed to refresh the gun.