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 $[\hbar]=\frac{q^\hbar-q^{-\hbar}}{q-q^{-1}}$. 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 are generally no known hopf algebra structures][1]  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*][2]" in the literature) to pull back the $q$-deformed fock spaces of the $q$-deformed oscillators to representations of the corresponding quantum group $U_q(g)$.  
Such methods have been applied since the '80's: [L. C. Biedenharn][3] and [A. J. Macfarlane][4] have provided descriptions of $su_q(2)$ deformations and their corresponding representations. A more complete account can be found in: [Quantum Group Symmetry and Q-tensor Algebras][5]. Similar methods have been used for the study of [$su_q(1,1)$][6] deformed lie algebra representations. Two parameter $(q,s)$-deformations have also been studied, see for example the case of [$sl_{q,s}(2)$][7] ... etc. The mathematical physics literature abounds of such examples during the last couple of decades. 

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: Such are the [Holstein-Primakoff][8], the [Dyson][9] or the [Schwinger][10] (see: [ch.3.8][9]) realizations (see also: [Dictionary on Lie Algebras and Superalgebras][11] or [this article][12] or [this one][13]).
If you are interested in such topics (in the deformed or the undeformed sense), i can provide further references (i have done some work on similar stuff during my phd thesis).  


  [1]: https://mathoverflow.net/a/259952/85967
  [2]: https://www.sciencedirect.com/science/article/pii/0370269390920043
  [3]: http://iopscience.iop.org/article/10.1088/0305-4470/22/18/004/pdf
  [4]: http://iopscience.iop.org/article/10.1088/0305-4470/22/21/020/pdf
  [5]: https://books.google.gr/books?id=DTlqDQAAQBAJ&pg=PA280&lpg=PA280&dq=L.%20C.%20Biedenharn,%20J.Phys.%20A:%20Math.%20Gen.%2022,%20L873-L878(19S9)&source=bl&ots=lnfGc_XHr1&sig=iJ6l7fi0XX_FEYBFyXqsYImDMtc&hl=el&sa=X&ved=0ahUKEwiMpbvxuMfYAhVFY1AKHVvQCnIQ6AEINDAC#v=onepage&q=L.%20C.%20Biedenharn%2C%20J.Phys.%20A%3A%20Math.%20Gen.%2022%2C%20L873-L878(19S9)&f=false
  [6]: https://link.springer.com/article/10.1007/BF00402678
  [7]: http://www.worldscientific.com/doi/abs/10.1142/S0217732393000568?journalCode=mpla
  [8]: https://en.wikipedia.org/wiki/Holstein%E2%80%93Primakoff_transformation
  [9]: https://books.google.gr/books?id=MID1lDaj9hgC&pg=PA67&lpg=PA67&dq=dyson%20boson%20realization&source=bl&ots=quGfv2mbrw&sig=cFkqYxBaoN3y0zMhcMTXX0tbKWw&hl=el&sa=X&ved=0ahUKEwiPtIqt7cnYAhWDJ1AKHTTnDXoQ6AEIPTAE#v=onepage&q=dyson%20boson%20realization&f=false
  [10]: http://www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf
  [11]: https://books.google.gr/books/about/Dictionary_on_Lie_Algebras_and_Superalge.html?id=HS7vAAAAMAAJ&redir_esc=y
  [12]: http://inspirehep.net/record/1388085/files/3-540-54040-7_109.pdf
  [13]: https://books.google.gr/books?id=sc1gDQAAQBAJ&pg=PA406&lpg=PA406&dq=realizations%20of%20q-deformed%20lie%20algebras&source=bl&ots=JwEqAh4pzb&sig=UX25D4HzEKt4SIdnPLgutzaUqxc&hl=el&sa=X&ved=0ahUKEwi9yoPllsTYAhUFPVAKHQe0BXMQ6AEIeDAJ#v=onepage&q=realizations%20of%20q-deformed%20lie%20algebras&f=false