In this paper we perform an exact study of ``Quantum Fidelity'' (also called
Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of
Hamiltonian : $$ \hat H\_{g}(t):=\frac{P^2}{2}+
f(t)\frac{Q^2}{2}+\frac{g^2}{Q^2} $$ when compared with the quantum evolution
induced by $\hat H\_{0}(t)$ ($g=0$), in the case where $f$ is a $T$-periodic
function and $g$ a real constant. The reference (initial) state is taken to be
an arbitrary ``generalized coherent state'' in the sense of Perelomov. We show
that, starting with a quadratic decrease in time in the neighborhood of $t=0$,
this quantum fidelity may recur to its initial value 1 at an infinite sequence
of times {$t\_{k}$}. We discuss the result when the classical motion induced by
Hamiltonian $\hat H\_{0}(t)$ is assumed to be stable versus unstable. A
beautiful relationship between the quantum and the classical fidelity is also
demonstrated.