We proposed a narrow spectral bandwidth Erbium-doped fiber (EDF) laser Q-switched by a homemade saturable dynamic induced grating (SDIG) which is introduced via reforming the structure of a fiber saturable absorbers FSA with a piece of EDF and a fiber Bragg grating. The SDIG integrates both saturable absorption and spectral filtering effect simultaneously, which was confirmed through theoretical analysis and experimental results for the first time, to the best of our knowledge. Further study verified that the spectral width of the Q-switched emissions is decided by the length of the SDIG and the input power of the pump source. The Q-switched pulse with the narrowest spectral width of about 29.1 pm achieved in this work is the narrowest bandwidth pulse in the domain of the FSA Q-switched fiber lasers when the length of SDIG and pump power are 20 cm and 250 mW, respectively. Our method provides a simple way to obtain the Q-switched pulses with narrow bandwidths, which have promising applications for nonlinear frequency conversion, Doppler LIDAR and coherent beam combinations.