I have a sink fixture in an apartment that I would like to connect to a filter. unscrewing the front of the fixture yields a hose terminating in a 3/8" diameter fitting. But it only has a TPI of 13 or 14 (not a perfect measurement). I am pretty baffled and not sure what I'm looking at. Does anyone know what kind of connection that is or where I can find an adapter?
Faucet spout aerator and hose threads come in many different sizes and threads. There is no real standardization between makes and models.
Some hose fittings may even be proprietary, as in deliberately not adaptable to other items. This is because assemblies with a hose/sprayer that could accidentally be dropped into dirty water often must be designed with an anti-siphon function. The manufacturer does not want the end-user to tamper with their assembly, perhaps defeating the anti-siphon feature.
Anyway, I strongly advise that you take your threaded parts to a good plumbing shop. There you will find a display with at least 15 different faucet spout thread sizes/threads and adapters to help mate them up.
There are small filters that can be installed on the end of the faucet made by Pur, Brita, and Culligan and others that have a charcoal filter and can be turned on/off when filtered drinking water is desired. You could also install a cartridge filter inside the cabinet under the sink and connected to the cold water line. Looking at the picture you posted I have no idea of of what you are asking. A better picture and description of what you want the end product to be or what you are seeking would provide better answers.