In this paper, two new families of MDS quantum convolutional codes are constructed. The first one can be regarded as a generalization of \cite[Theorem 6.5]{GGGlinear}, in the sense that we do not assume that $q\equiv1\pmod{4}$. More specifically, we obtain two classes of MDS quantum convolutional codes with parameters: {\rm (i)}~ $[(q^2+1, q^2-4i+3,1;2,2i+2)]_q$, where $q\geq5$ is an odd prime power and $2\leq i\leq(q-1)/2$; {\rm (ii)}~ $[(\frac{q^2+1}{10},\frac{q^2+1}{10}-4i,1;2,2i+3)]_q$, where $q$ is an odd prime power with the form $q=10m+3$ or $10m+7$ ($m\geq2$), and $2\leq i\leq2m-1$.