Does gray coding bring improvement to SER of QPSK, compared with natural coding?
The central point as crystallized in the comments is that you think SER is derived from BER.

It's not! 

SER is short for "symbol error rate", and that's it. That's the underlying thing.

A symbol error is when one symbol was sent, and a different one was received. It doesn't matter whether the symbol you've sent was "labeled" 00, or 11, or 10; or whether the received symbol was defined to be 00, 11 or 10, or yoghurt for that matter; what matters is that not the symbol that the sender sent was decided for in the receiver.

Thus, the SER doesn't care about what binary values are assigned to the symbols. (and that assignment can be Gray, or something else.)

By the way, I'm not sure what you mean with "natural encoding": I know that term from one-dimensional encoding problems (e.g. ADC values), where natural is just binary counting up:

value    |  0  1  2  3
---------+----------------
encoding | 00 01 10 11


If you extend it two a second dimension by just forming coordinate vectors, you get gray encoding for QPSK:

    \ dim0  0   1
      -------------
dim1 |
     |
   0 |     00  01
     |
   1 |     10  11


So, I'd argue Gray and natural are the same here, and using anything different would just be ill-advised .