  In this paper we give first examples of $\mathbb{Q}$-Fano threefolds whose
birational Mori fiber structures consist of exactly three $\mathbb{Q}$-Fano
threefolds. These examples are constructed as weighted hypersurfaces in a
specific weighted projective space. We also observe that the number of
birational Mori fiber structures does not behave upper semi-continuously in a
family of $\mathbb{Q}$-Fano threefolds.
