Understanding the origin of the magnetism of high temperature superconductors is crucial for establishing their unconventional pairing mechanism. Recently, theory predicts that FeSe is close to a magnetic quantum critical point, and thus weak perturbations such as impurities could induce local magnetic moments. To elucidate such quantum instability, we have employed scanning tunneling microscopy and spectroscopy. In particular, we have grown FeSe film on superconducting Pb(111) using molecular beam epitaxy and investigated magnetic excitation caused by impurities in the proximityinduced superconducting gap of FeSe. Our study provides deep insight into the origin of the magnetic ordering of FeSe by showing the way local magnetic moments develop in response to impurities near the magnetic quantum critical point.