  Using the notion of quantum integers associated with a complex number $q\neq
0$, we define the quantum Hilbert matrix and various extensions. They are
Hankel matrices corresponding to certain little $q$-Jacobi polynomials when
$|q|<1$, and for the special value $q=(1-\sqrt{5})/(1+\sqrt{5})$ they are
closely related to Hankel matrices of reciprocal Fibonacci numbers called
Filbert matrices. We find a formula for the entries of the inverse quantum
Hilbert matrix.
