The functions $$ C_n(q)=\sum_{P\in\square_n}q^{area(P)} $$ satisfy the following recurrence relation $$ C_n(q)=\sum_{k=1}^nq^{k-1}C_{k-1}(q)C_{n-k}(q).\tag{1} $$ Proof. (taken from the book "The q, t-Catalan Numbers and the Space of Diagonal Harmonics" by J. Haglund"The q, t-Catalan Numbers and the Space of Diagonal Harmonics" by J. Haglund, page 14, typo corrected)
We break up our path $P$ according to the “point of first return” to the line y = x. If this occurs at (k, k), then the area of the part of $P$ from (0, 1) to (k − 1, k), when viewed as an element of $\square_{k-1}$, is $k − 1$ less than the area of this portion of $P$ when viewed as a path in $\square_{n}$.$\qquad\qquad\qquad~~~~~\Box$
The generating function $$ G(z,q)=\sum_{n=0}^\infty C_n(q)z^n $$ satisfies the functional equation, which is a direct consequence of $(1)$: $$ G(z,q)-1=zG(z,q)G(qz,q). $$ We rewrite it for convenience as $$ \frac{1}{G(z,q)}=1-zG(qz,q).\tag{2} $$ It is known that the generalized Rogers-Ramanijan continued fraction, formula (39), $$ F(a,q)=\frac{\sum_{k=0}^\infty\frac{(-a)^kq^{k^2}}{(q;q)_k}}{\sum_{k=0}^\infty\frac{(-a)^kq^{k^2+k}}{(q;q)_k}}=1-\cfrac{aq}{1-\cfrac{aq^2}{1-\cfrac{aq^3}{1-...}}}, $$ satisfies the functional equation $$ F(a,q)=1-\frac{aq}{F(aq,q)}.\tag{3} $$ Since $G(0,q)=F(0,q)=1$ it is clear from comparing $(2)$ and $(3)$ that $$ \frac{1}{G(z,q)}=F(z/q,q)=\frac{\sum_{k=0}^\infty\frac{(-z)^kq^{k^2-k}}{(q;q)_k}}{\sum_{k=0}^\infty\frac{(-z)^kq^{k^2}}{(q;q)_k}}, $$ as required.
