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Quarter turn metric is more difficult to handle than h-w metric, because the 180 degree turn has to be counted as two moves, which gives some issues with an recursive approach. I did not believe it was possible to get a simple formula here. I was very surprised, that the result was a simple generating function for the number of canonical sequences in q-w-metric. It is

gfq[n_,x_]:=3/(6-4(x+1)^(2(n-1)))-1/2

and looks very similar to the generating function in h-w metric which is

gfh[n_,x_]:= 3/(6-4(3x+1)^(n-1))-1/2