It's not hard to compute numerical values. If you do this, in the regime $0 < q < 1$ it looks like $C_n$ grows exponentially, i. e. $C_n sim alpha_q beta_q^n$ for some constants $alpha_q$ and $beta_q$ which depend on q.
Unfortunately, I don't know what $alpha_q$ and $beta_q$ are. For example, when q = 1/2 the ratio $C_n/C_{n-1}$ approaches a constant which is approximately 1.6022827223; I claim this is $beta_{1/2}$. Then $C_{50}/beta_{1/2}^{50} = 0.5757566503$, which I claim is $alpha_{1/2}$. Neither of these constants appears in the inverse symbolic calculator.
The generating function $C(q,z) = C_0 + C_1 z + C_2 z^2 + ldots$, where the $C_n$ are $q$-Catalan numbers, ought to satisfy some functional equation, and then one could use techniques from singularity analysis (see, for example, Analytic Combinatorics by Flajolet and Sedgewick). But I am having trouble finding that functional equation.
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