The aim of this paper is to obtain quasimodes for a Schrödinger type operator P _h in a semi-classical limit $ (h \searrow 0) $ with exponentially small error terms which are associated with Gevrey families of KAM tori of its principal symbol H. To do this we construct a Gevrey quantum Birkhoff normal form of P _h around the union $ \Lambda $ of the KAM tori starting from a suitable Birkhoff normal form of H around $ \Lambda $ . As an application we prove sharp lower bounds for the number of resonances of P _h defined by complex scaling which are exponentially close to the real axis. Applications to the discrete spectrum are also obtained.