In this paper, a tunable terahertz (THz) absorber operating at a quasi-bound state in the continuum (quasi-BIC) mode supported by a graphene metasurface is proposed. There are two graphene strips and a fully covered graphene layer in one unit cell. By breaking the symmetrical arrangement of graphene stripes, the symmetry-protected BIC transforms into a quasi-BIC mode. The reflective configuration results in high-Q absorption of the metadevice at the quasi-BIC mode with the equivalent impedance matching the impedance in free space. The change in the Fermi level of graphene can cause a frequency shift in the position of the absorption peak at the quasi-BIC mode. Benefiting from the high Q-value and narrow linewidth of the quasi-BIC, the frequency shift of the absorption peak can easily exceed its linewidth. At this time, the designed THz absorber can be used as a switch, and the “on” and “off” states are achieved by tuning the Fermi level of graphene. Under normal incidence, the modulation depth of the absorption type THz switch can reach up to 99% with the insertion loss only 0.062 dB. Within the range of incident angle inclination approaching 10°, the absorption type THz switch can still achieve more than 90% modulation depth and insertion loss below 0.1 dB. Due to the characteristics of large modulation depth, low insertion loss, and wide angle incidence, the designed tunable THz absorber has great application prospects in fields such as THz communication and THz wavelength division multiplexing.