Our friend Fran Kent gives me rather too much credit in referring to me as "a master of web research" but I will take any accolade that comes my way. Thanks, Fran.
By way of caveat, I am the webmaster of the U.S. Navy Armed Guard / U.S. Merchant Marine website. This website is concerned with the Navy Armed Guard, a branch of the Navy that, during World War II, provided personnel to man defensive weapons, and to provide communications, on merchant ships (supply ships). As such the website has nothing to do with naval aviation; personally I would claim to be a "master" (per Fran) only on researching information about the Armed Guard and the merchant marine. So expect no miracles from me.
Nevertheless, I'll take a try at your inquiry.
"NTS San Diego" refers to the naval training station located in San Diego. Your father may have taken initial or advanced training there but it was probably not his duty station for any extended period of time.
As Fran notes, "Hedron Fair Wing Two" means headquarters squadron, fleet air wing two. From what I can find, and to confirm Fran's statement, this unit was based in Hawaii, originally at Pearl Harbor and later at NAS (Naval Air Station) Kaneohe Bay, apparently for the entire war. The facility is now a Marine Corps aviation base.
Since the unit was stationed on land, it seems entirely possible that your father was never assigned to a ship but carried out his duties shore-side. The unit apparently was involved in the operation, maintenance and repair of aircraft that, to be sure, were deployed to aircraft carriers but which were also sent to the air station as necessary for maintenance. I admit I'm speculating here about your father's specific assignments. I know, I know, if your father was in the Navy you would expect him to be on a ship but ... maybe not.
I have had some success in Googling "fair wing 2" "fair wing two" "fleet air wing 2" "fleet air wing two" and "FAW-2" to find information about this unit. See, for example, http://www.vpnavy.com/faw2.html, which has a number of links that may or may not prove useful. Also see http://www.vpnavy.com/guestbook.html and http://www.vpnavy.com/forum.html. It looks like one can post photographs on this website but I don't know the procedure. Contact the webmaster of the site at email@example.com.
Also see http://www.history.navy.mil/avh-vol2/Appen13.pdf for a detailed history of fleet air wings.
There is a lot of information to be pursued in these searches. Whether any will be of use to you is something only you can determine.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website