Will iTunes recognize my iPhone 5 from a iPhone 4 adapter?

I want to know if that iPhone 4 to iPhone 5 adapter will work when I go to plug it into my computer to put songs only phone?

No it won't because the iphone 4 has a 30 pin charger and the newer models have a lightning charger.

The computer does not care about adapters. As long as the adapter is functioning properly your device will be recognized.