Let us use the standard notation for $q$-integers, $q$-binomials,
and the $q$-analog 
$$
\operatorname{Cat}\_q(n) := \frac{1}{[n+1]\_q} \left[\matrix{2n \\ n}\right]\_q.
$$
I want to prove that for all integers $n\geq 0$, we have
\begin{equation}
\operatorname{Cat}\_q(n+2)
=
\sum\_{0\leq j,k \leq n}
q^{k(k+2) + j(n+2)}
\left[\matrix{n \\ 2k}\right]\_q
\operatorname{Cat}\_q(k)
\frac{[n+4]\_q}{[k+2]\_q}
\left[\matrix{n-2k \\ j}\right]\_q.
\end{equation}


I have tried quite a bit, but not succeeded. Using $q$-hypergeometric series,
this is equivalent with proving
$$
\sum\_{\substack{k\geq 0 \\ j \geq 0}}
q^{k(k+2)+j(n+2)}
\frac{
(q;q)\_{n+4}
}{
(q^{n+3};q)\_{n+2} (q;q)\_{j}
}
\frac{
(q^{n-2k+1};q)\_{2k}
(q^{n-2k-j+1};q)\_{j}
}{
(q;q)\_{k}
(q;q)\_{k+2}
}
=1
$$
which I have also not managed to prove.
I believe that some WZ-method could solve this easily,
but a human-friendly proof would be preferrable. 
Note that the identity above is very similar to a 
theorem by Andrews (see reference below). It states that
\begin{equation}
\operatorname{Cat}\_q(n+1)
=
\sum\_{k \geq 0}
q^{k(k+2)}
\left[\matrix{n \\ 2k}\right]\_q
\operatorname{Cat}\_q(k)
\frac{(-q^{k+2};q)\_{n-k}}{(-q;q)\_k}.
\end{equation}


UPDATE: I have managed to find a more 
general conjecture, which would imply the one above.
It states that for integer $n \geq 0$, and general $a,c$,
we have
$$
 \sum\_{s}
 \frac{ (-a q^n)^{s}
q^{-\binom{s}{2}}
(q^{-n};q)\_{s} }{ (q;q)\_{s} } 
{}\_{2}\phi\_{1}(cq^{s-1}/a,q^{-s};c;q,q)
=
\frac{ (ac ;q)\_{n} }{ (c;q)\_{n} }.
$$


I use [This book](http://www.fuchs-braun.com/media/e4ef05831b027c9cffff8033ffffffef.pdf) as my main reference for notation and identities.


*Andrews, George E.*, [**(q)-Catalan identities**](http://dx.doi.org/10.1007/978-1-4419-6263-8_10), Alladi, Krishnaswami (ed.) et al., The legacy of Alladi Ramakrishnan in the mathematical sciences. New York, NY: Springer (ISBN 978-1-4419-6262-1/hbk; 978-1-4419-6263-8/ebook). 183-190 (2010). [ZBL1322.11018](https://zbmath.org/?q=an:1322.11018).



Proof of certain $q$-identity for $q$-Catalan numbers
I managed to solve the problem,
in the last general conjecture, one can apply the $q$-Chu-Vandermonde theorem. After some simplification,
the resulting expression can be expressed as a ${}\_2\phi\_1$ q-hypergeometric series, where one again can apply the $q$-Chu-Vandermonde theorem.


Skipping lots of details the proof still requires a few pages - it can now be found [in this paper](https://arxiv.org/abs/2010.11157) (Thm. 57)


