How to plot BER Vs SNR of a received signal
From the plot it seems the transmitted QAM points are $(0.5, 0.5),(-0.5,0.5),(-0.5,-0.5),(0.5,-0.5)$, and if you can model your comm system as just AWGN channel, then your signal amplitude will just be $\sqrt{0.25+0.25} = \sqrt{0.5}$ and hence signal power will be $0.5$. In doing so, I have assumed all 4 QAM symbols equiprobable. 

You can get your noise power by averaging all squared Euclidean distances of received points from their true QAM points.

From this you can get received SNR.