Is there a geometry in which $\pi$ is a natural number? Is there a geometry in which $\pi$, the ratio of a circle's circumference to its diameter, is an integral number such as $3$ or $4$?
If you use the distance metric of $d=|x_1-x_2|+|y_1-y_2|$ rather than the Pythagorean one then a circle (all points equal distance from a centre) looks like a diamond and the ratio of circumference to diameter is 4.



Interestingly additional fact if you consider all related distance metrics: $d_n=\left(|x_1-x_2|^n+|y_1-y_2|^n\right)^{\frac1n}$ and draw the corresponding circle and then measure the circumference (non-trivial due to different distance metric) and hence calculate $\pi_n$ that the value we use for $\pi$ (in this case $\pi_2$) is a minimum value for $\pi_n$.
In any geometry, in which circles can be defined, either the ratio of circumference of a circle to its diameter is $\pi$ or that ratio is not a constant.  So, no, there is no geometry in which that ratio is constant other than $\pi$.