In your notation $k$ is the degree and $q$ the component of the tensor. You can easily verify that $z$ is proportional to the $0$th component since the components satisfy $[L_z,T^{(k)}_q]=q T^{(k)}_q$. Moreover $z\sim Y_{10}(\theta, \phi)$, confirming the value $q=0$ for the component.

In your notation $k$ is the degree and $q$ the component of the tensor. You can easily verify that $z$ is proportional to the $0$th component since the components satisfy $[L_z,T^{(k)}_q]=q T^{(k)}_q$. Moreover $z\sim Y_{10}(\theta, \phi)$, confirming the value $q=0$ for the component (and for that matter that $k=1$).