I don't believe this is possible to get in a clean manner. The protocols - a, b, g, n, etc - are actually human abstractions of the MAC and physical layer in networks. These are defined as their recognizable letters if they meet certain hardware specifications, both for the device and the network device.
After doing some digging, it seems that Windows phones are able to display this information. When digging into the manner of determining the protocol on Windows, I came across the desktop explanation as well as the Visual C++ implementation via enums. It seems that even the official Windows documentation relies on vendor-provided data and enumerated values, which would lead me to believe that they need to determine hardware specifications beyond what is likely exposed in the Android API.
If you want to determine the protocol yourself, you'll need to understand the link speed as well as the frequency, modulation, and bandwidth.
TL;DR
Likely not possible unless you are working with a rooted phone and can access the hardware specs directly.