BER of QPSK modulation in Rayleigh fading channel model
Chapter 14 of John Proakis' Digital Communications would be helpful, and Appendix C of the same might actually do that derivation. At the least, Proakis gives the expression for M-ary PSK in the Rayleigh channel which you could reduce for 4-PSK.

I remember it by remembering the basic AWGN channel. Since the Q function is the tail of a Gaussian distribution, BER of QPSK in AWGN is found using SNR and the Q function. Rayleigh distribution can be thought of as a complex valued AWGN. One of the easiest ways to derive Rayleigh is to start with two AWGN PDFs that are orthogonal. So I'm pretty sure you just use the SNR and the CDF called Rayleigh distribution.