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The paper by Cherednik On q-analogues of Riemann's zeta function gives precisely the definition you're after: $$\zeta_q(s)=\sum\limits_{n=1}^\infty q^{sn}/[n]_q^s$$ This His paper also contains a brief discussions discussion of the properties of this $q$-zeta function. The On the other hand, the term quantum zeta function appears to have a somewhat different meaning, see e.g. this the paper On the quantum zeta function by R.E. Crandall.
The paper by Cherednik On q-analogues of Riemann's zeta function gives precisely the definition you're after: $$\zeta_q(s)=\sum\limits_{n=1}^\infty q^{sn}/[n]_q^s$$ This paper also contains a brief discussions of the properties of this $q$-zeta function. The term quantum zeta function appears to have a somewhat different meaning, see e.g. this paper.