I think that a modern realistic perception of the term "quantum algebra" has to be understood in its historical context, that is the study of the quantization problem in its various forms (first and second quantization , QFTs etc):
As far as i know, the term quantum algebra, has been introduced in Dirac's seminal paper "The fundamental equations of quantum mechanics", Proc. Roy. Soc. A, v.109, p.642-653, 1925 (a reprint can be found in Sources of Quantum Mechanics, ed. B.L. van der Waerden, p.307). It was short after Heisenberg had proposed his -revolutionary for the time- idea that quantum observables should correspond to hermitian matrices of -generally- infinite order. However, he considered the non-commutativity of the matrices as an obstacle in the further development of the idea. Heisenberg communicated his ideas to Fowler at Cambridge. Fowler, was by that time, the thesis advisor of Dirac and this is how the latter got involved. Dirac shortly proposed that the non-commutativity of quantum mechanical observables should be treated as a fundamental characteristic of the new theory to be developed. He also proposed that quantum observables $A$ and $B$ should belong in a non-commutative algebra, satisfying the relation
$$
[A,B]=i\hbar \{A,B\}
$$
as a "measure of departure" from commutativity. ($[.,.]$ stands for the commutator and $\{.,.\}$ for the classical Poisson bracket). A detailed account of the history of the development of the notion of quantum algebra together with references, historical and technical details, can be found at Varadarajan's Reflections on Quanta, Symmetries and Supersymmetries.
During the next decades, the term quantum algebra started expanding and embracing new ideas and methods emerging from the studies of different aspects of the various quantization problems. Dirac's commutator was replaced by Moyal bracket (coinciding with Dirac's comm. modulo $\hbar^2$ terms) and this is how the deformation theory (already developed as a separate discipline at the level of assoc. and Lie algebras) entered the picture. Now quantum mechanical algebras of observables were viewed as deformations of the corresponding classical objects. Moshe Flato and his coworkers have been among the pioneers at that direction.
The rise of quantum groups and $q$-mathematics, expanded the term even more. Now whole new families of examples and methods arose, introducing new mathematical ideas and tools into the subject, such as hopf algebras, $q$-analytical tools, representation-theoretic methods, $q$-deformations of Weyl algebras etc.
The continuous development of Quantum Field theories together with the various technical and conceptual problems introduced by them, led to further expansions of the discipline of quantum algebras. Now, algebraic geometric, homological, homotopical and Category theoretical methods and notions got involved. The development of non-commutative geometry, also opened new directions of study. I am far from being an expert into such topics to provide further details but i have the feeling that almost everything inside "quantum algebras" has been in some way connected or at least originating (even in some distant sense) from the study of the quantization problems.
So, concluding, i would say that, although the requirement for a one-sentence definition of the topic of quantum algebra might seem superficial, a rough approximation (modulo my understanding of course) might be:
The study of the algebraic/geometric theories, methods, techniques, notions and questions originating from the study of the various aspects of the quantization problems (broadly interpreted).