Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,

$$\pi_1^{orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$

$$\pi_1^{orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$

Want to know the centre of these two groups. Are they trivial?