Is there a "quantum" Riemann zeta function?

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

Here is another article dealing with similar functions:

q-analogue of Riemann’s ζ-function and q-Euler numbers. by Junya Satoh.

There are also many articles by Taekyun Kim on related functions.

One key point is that the value of the function $\zeta_q$ at negative integers is a fraction which has no limit when $q$ goes to $1$. One can obtain a relation to the $q$-Bernoulli numbers introduced by Carlitz in 1948, by taking a difference with the value of a modified $\zeta_q$ function.