Motzkin numbers are a very popular sequence. A lot of identities and formulas are already recorded at OEIS. The analogous integral representation for Motzkin numbers is
$$M_n=\frac{1}{2\pi} \int_{-1}^3 x^n\sqrt{(3-x)(1+x)}dx.$$

A few words about the general picture. There is a well known combinatorial theory of orthogonal polynomials and continued fraction which is closely related to Motzkin paths.

By Favard's theorem we know that a sequence of polynomials is a system of orthogonal polynomials with respect to some measure if and only if they satisfy a recurrence for all $n$
$$P_n=(x-a_n)P_{n-1}-b_nP_{n-2}$$
Now the generating functions of moments $\sum \mu_ix^i$ when written as a continued fraction tells us the coefficients $a_n,b_n$ and therefore the sequence of orthogonal polynomials. One can also interpret these continued fractions as generating functions for weighted Motzkin paths. This is a theorem of Viennot

The moments of a sequence of orthogonal polynomials are sums of Motzkin paths of length $n$ where horizontal steps at height $k$ are weighted by $a_k$ and down steps at height $k$ are weighted by $b_k$.

As a special case the numbers $M_n$ appear as moments of the sequence of polynomials which satisfies $P_n=(x-1)P_{n-1}-P_{n-2}$ which are shifted and scaled Chebyshev polynomials of second kind, and the identity above holds. However one can give such a combinatorial context to most known families of orthogonal polynomials.

For the relation between orthogonal polynomials and continued fractions, I would recommend "Orthogonal polynomials and random matrices: a Riemann-Hilbert approach" by P. Deift, it is short and a very enjoyable read. For the combinatorial theory see for example "Combinatorial aspects of continued fractions" by P. Flajolet or Viennot's works on orthogonal polynomials which you can find here.