I found an adapter and now the issue resolved.

Hello, I have the same issue as yours, and last night I tried 1/8 FIP-1/4 FIP; however, 1/8 inch female end is not coupling with the 1/8 inch flare on the back-flow unit. Can I please know what specific adapter you used? Thank you.

one thing you both may of missed is that the 1/8" male fitting unscrews from the test cock and a standard 1/4" air hose fitting will thread into the test cock.

Thank you, MrHIrrigation, eventually I took the 1/8" male fitting out and used a 3/8" to 1/4" coupler with that. (the test cock remaining is a 3/8" female).

Four years later, In case anyone else stumbles upon this, here's some more helpful details vs. "I found an adapter". The test cock threads on the Febco 765 are "flared". I'm not a plumber but I was told the intent by Febco is to use flared threads to discourage direct air compressor hookups to the test cocks even though 90% of people use them for this purpose. So, it takes a special adapter to screw on the threads of the test cock. If you go the route I did and first try to fit a $4 1/8 inch female adapter from Lowe's, you'll quickly notice that the threads don't fit (like a previous responder wrote). The part I found is called "1/4 Inch Male Pipe X 1/4 inch Female Flare Brass Adapter". I found it at an irrigation store for $7 so I'm unsure if it is available at the big box Home stores. Hope this helps the next person that is very confused by threading issues on the Febco 765. Good luck.