We study the $L^q$ dimension $D(ν,q)\ (q>1)$ of stationary measures $ν$ for Möbius iterated function systems on $\mathbb{R}$ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]{Shm19}. As the result, we show that there is the dichotomy: the $L^q$ spectrum $τ(ν,q)=(q-1)D(ν,q)$ is equal to the desired value $\min\{\widetildeτ(ν,q),q-1\}$ for any $q>1$, where $\widetildeτ(ν,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<α<1$ such that $τ(ν,q)=\min\{\widetildeτ(ν,q),q-1\}$ for $1<q<q_0$ and $τ(ν,q)=αq$ for $q\geq q_0$. In addition, we give examples of Möbius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.