If i have correctly understood your question, there are various such examples, arising from mathematical physics contexts (where some of the original motivations for the study of quantum groups first appeared).
One such example, is the case of the $q$-Heisenberg algebra: If we consider the (usual) 3d Heisenberg Lie algebra $L_H$, generated by $a,a^{\dagger},H$ subject to the relations:
$$
[a,a^\dagger]=H, \ \ \ \ \ \ \ \ \ [H,a]=[H,a^\dagger]=0
$$
then the $q$-deformed Heisenberg algebra (with $q$ a non-zero parameter), may be defined in terms of generators $a,a^\dagger,q^\frac{H}{2},q^{-\frac{H}{2}}$ and $1$ and relations:
$$
q^{\pm\frac{H}{2}}q^{\mp\frac{H}{2}}=1, \ \ \ \ \ [q^\frac{H}{2},a]=[q^\frac{H}{2},a^\dagger]=0, \ \ \ \ \ [a,a^\dagger]=\frac{q^H-q^{-H}}{q-q^{-1}}
$$
(Of course now $[.,.]$ is no more the Lie bracket but simply the usual commutator). This is known to be a quasitriangular hopf algebra. It may be thought of, as the $q$-deformation $U_q(L_H)$ of the universal enveloping algebra $U(L_H)$ of the Heisenberg Lie algebra $L_H$.
It can be shown, that, if $q$ is a real number, then the unitary representations of $U_q(L_H)$ are parameterized by a real, positive parameter $\hbar$. If we denote the basis vectors by $$H_\hbar=\{|n,\hbar\rangle\big{|}n=0,1,2,...\}$$
then the action of the generators is given by:
$$
|n,\hbar\rangle=\frac{(a^\dagger)^n}{[\hbar]^{\frac{n}{2}}\sqrt{n!}}|0,\hbar\rangle, \ \ \ \ \ q^{\pm\frac{H}{2}}|0,\hbar\rangle=q^{\pm\frac{\hbar}{2}}|0,\hbar\rangle, \ \ \ \ \ a|0,\hbar\rangle=0
$$
where $[\hbar]=\frac{q^h-q^{-h}}{q-q^{-1}}$. This a deformation of the usual Fock representation of the Heisenberg Lie algebra. If you are interested in similar examples, you can find more in S. Majid's book, "Foundations of Quantum group theory".
Furthermore, various $q$-deformations of the harmonic oscillator algebra can be used for a more systematic way of constructing such examples: Although most of $q$-deformed CCR are not quantum groups themselves (up to my knowledge there is no known hopf algebra structure for such algebras), suitable homomorphisms from quantum groups $U_q(g)$ (with $g$ being any Lie (super)algebra) to $q$-deformations of the harmonic oscillator can be used (such homomorphisms are usually called "realizations" in the literature) to pull back the q-deformed fock spaces to representations of the corresponding quantum group. See for example this article or this one.
The situation is similar to the way, various bosonic or fermionic realizations of Lie (super)algebras have been used to construct Lie (super)algebra representations, initiating from the usual symmetric/antisymmetric bosonic/fermionic Fock spaces.
If you are interested in such topics, i can provide further references (i have done some work on similar stuff during my phd thesis).