  We obtain a new $q$-analogue of the classical Leibniz series
$\sum_{k=0}^\infty(-1)^k/(2k+1)=\pi/4$, namely \begin{equation*}
\sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}},
\end{equation*} where $q$ is a complex number with $|q|<1$. We also show that
the Zeilberger-type series
$\sum_{k=1}^\infty(3k-1)16^k/(k\binom{2k}k)^3=\pi^2/2$ has two $q$-analogues
with $|q|<1$, one of which is $$\sum_{n=0}^\infty q^{n(n+1)/2} \frac
{1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2
\frac{(q^2;q^2)^4_\infty}{(q;q^2)^4_\infty}.$$
