Although the design of graphene-based tunable broadband terahertz (THz) absorbers has attracted much attention, improving the functionality of the absorbers to adapt to different scenarios is still worth studying. This paper presents an innovative design of a quad-functional metasurface absorber (QMA) in the THz region, which can switch the absorption frequency/band by means of dual voltage/thermal manipulation. By electrically manipulating the chemical potential of graphene, the QMA can switch freely between the narrowband absorption mode ("NAM") and the broadband absorption mode ("BAM"), while thermally manipulating the phase transition of VO2 allows switching between the low-frequency absorption mode ("LAM") and the high-frequency absorption mode ("HAM"). Detailed mechanistic analysis shows that the "NAM" and "BAM" are due to the switching of the fundamental and second order graphene surface plasmon polariton (SPP) resonances, respectively, and the switching between "LAM" and "HAM" is due to the phase transformation of VO2. Furthermore, the QMA is polarization insensitive in all absorption modes and maintains excellent absorption performance at large angular incidence of TE- and TM-polarized waves. All the results indicate that the proposed QMA has great potential for stealth, sensing, switching, and filtering applications.