Thanks, Pei-Ming. Gee get a letter from him. The following is his interpretation of a quantum torus:

First let me explain the general idea of a quantum space.
The mathematician A. Connes has a book
"Noncommutative Geometry" and it is supposedly
one of the origins of quantum spaces like quantum
torus. The general idea is this. Since many properties
of a classical space (a manifold or a topological space)
can be described as properties of the algebra of
functions on the space, we can try to define spaces
by talking about the algebra of functions only, without
talking about the points on the space. Then we
consider the generalization of the definition of spaces
(in terms of the algebra of functions) by allowing
the algebra (of functions) to be noncommutative.

The quantum torus is an example. It is defined only
in terms of the algebra of "functions on the quantum
torus", without talking about "points" on the space.
The algebra is defined as the linear space
{ U^m V^n } for all integers m, n.
U and V are the generators of this algebra.
They satisfy the following properties
U^* = U^{-1}, V^* = V^{-1}
(* = Hermitian conjugate).
UV = qVU, where q is a phase factor.
A different choice of the constant q gives
a different quantum torus.

If we take q = 1, then we get an ordinary torus
(we usually call it a "classical torus"). In that case,
U = e^{ix_1}, V = e^{ix_2}, and x_1, x_2 are
the coordinates on the torus and
x_1 ~ x_1 + 2\pi, x_2 ~ x_2 + 2\pi.

So this is my explanation.