There are several ways to express the quarter period $K$, $$ K(m)=\int_0^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^2\theta}}, $$ as a power series (and thus for $K'=K(1-m)$ there are, too) and also efficient ways of calculation (like agm). But are there any known formulas for the quotient $$ K'/K $$ ? I'm new to the field of elliptic functions.
Any ideas, general results etc.?
Here is what Whittaker and Watson tell us in their classic text A Course of Modern Analysis.
The parameter $m$ in your question is usually denoted by $k^2$ so that $k\in(0,1)$ and $k=\sqrt{m}$ and further let $k'=\sqrt{1-m}$. Set $$2h=\frac{1-\sqrt{k'}}{1+\sqrt{k'}}\tag{1}$$ so that $0<h<1/2$ and then we have the series $$q=h+2h^5+15h^9+150h^{13}+O(h^{17})\tag {2}$$ The ratio $K'/K$ is now given as $$\frac{K'} {K} =-\frac{1}{\pi}\log q\tag{3}$$