Is there any published physics article where $q$-mathematics is applied? Excuse me for the concern, but I want to ask you a question.
In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science of physics (see [1]). The scripts were interesting but so far I could not find any article where $q$-mathematics was applied. Is there any published article where $q$-mathematics is applied?
[1] http://math.ucr.edu/home/baez/week183.html
 A: There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformations of simple harmonic oscillator(s) and angular momentum algebras to the development of quantum groups and their applications in nuclear physics, particle physics and  field theories. They can be -roughly- divided in two broad categories (although experts might argue that such  classifications can be made much more fine):


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*Applications of phenomenological nature:
Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still lingers -see p.179- and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example 9 where the dark energy is essentialy considered to be a $q$-deformed scalar field.
Just to mention a few papers in this category (my former phd advisor has quite some work on the field):  


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*Generalized deformed oscillator and nonlinear algebras, C Daskaloyannis 1991 J. Phys. A: Math. Gen. 24 L789 

*Coupled $Q$-oscillators as a model for vibrations of polyatomic molecules, D.Bonatsos, C.Daskaloyannis, The Journal of Chemical Physics 106, 605 (1997)

*Quantum groups and their applications in nuclear physics, D.Bonatsos, C.Daskaloyannis, Progress in Particle and Nuclear Physics, v.43, 1999, p. 537-618 (see also here for the arxiv version). 

*The many-body problem for $q$-oscillators, E G Floratos, Journal of Physics A: Mathematical and General, Volume 24, Number 20, 1991 

*Dynamical algebra of the $q$‐deformed three‐dimensional oscillator, J. Van der Jeugt, J. of Math. Phys. 34, 1799 (1993)

*WKB equivalent potentials for the $q$-deformed harmonic oscillator, 
D Bonatsos, C Daskaloyannis and K Kokkotas, J. of Phys. A: Math. and Gen., Volume 24, Number 15, 1991

*Introduction to Quantum Algebras, Maurice R. Kibler, arXiv:hep-th/9409012 (see especially the discussion in sections 7,8 and the references). 

*An Introduction to Quantum Algebras and Their Applications, R. Jaganathan, arXiv:math-ph/0003018 (see the discussion of p.11-13)

*Interacting Dark Matter and $q$-Deformed Dark Energy Nonminimally Coupled to Gravity, Emre Dil, Advances in High Energy Physics, Volume 2016 (2016), Article ID 7380372  


*Applications of a more conceptual or fundamental nature:
-->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example this paper, this paper and this one (among lots of works in a similar spirit). The relation of $q$-mathematics and quantum groups to deformation quantization is another hot topic with a significant number -imo- of open questions. In Minimal areas from q-deformed oscillator algebras the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied.
-->The quantum inverse scattering method and the definition(s) and study of quantum integrability (see p. 269) have given rise to quantum groups and quantum algebras.  The mathematical developments associated with them have been greatly inspired and have -in return- significantly contributed to such directions of study. Jule Lamers' answer provides further details  and references on that point. S. Majid's book is also worth to be mentioned as it contains a cataclysm of such ideas and applications.
-->The case of deformed particles (deformed bosons or fermions) which interpolate between different statistics has been another line of interesting applications of this kind. 
A significant amount of related literature can be found at mathematical physics journals such as the Journal of Mathematical Physics, Journal of Physics A: Mathematical and General, Communications of Mathematical Physics, SIGMA etc. 
A: As another example of the second category in Kostantinos Kanakoglou's answer I think it is fair to mention quantum-integrable systems: this topic in physics was pivotal in the historical development of the notion of quantum groups by the Leningrad group (Faddeev et al), and the Japanese group (Jimbo and Miwa et al).
In particular, $U_q(\widehat{\mathfrak{sl}_2})$, the quantum-affine version of $\mathfrak{sl}_2$, naturally arises in the (algebraic Bethe-ansatz) analysis of the partially isotropic (or "XXZ") Heisenberg quantum spin chain, and the closely related six-vertex model in statistical physics. Here the deformation parameter $q$ characterizes the spin chain's partial anisotropy as $\Delta = (q+q^{-1})/2$, with $\Delta=q=1$ the completely isotropic ($\mathfrak{sl}_2$-invariant) point. 
More, including many references to published articles, can e.g. be found in the books 


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*Jimbo and Miwa, Algebraic Analysis of Solvable Lattice Models, esp Chapters 0--3;   

*Gómez, Ruiz-Altaba and Sierra, Quantum Groups in Two-Dimensional Physics, esp Section 2.3 and its references;

*Chari and Pressley, A Guide to Quantum Groups, esp Section 7.5 and its bibliographical notes. 


Finally, although I am by no means an expert on the topic, let me point out that the second half of the book by Gómez et al is devoted to quantum groups in conformal field theory, see also 


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*Fuchs, Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory.

