Simultaneous broadband absorption and polarization conversion are crucial in many practical applications, especially in terahertz communications. Thus, actively tunable metamaterial systems can exploit the graphene-based nanomaterials derived from renewable resources because of the flexible surface conductivity and selective permeability of the nanomaterials at terahertz frequencies. In this paper, we propose a graphene-based active tunable bifunctional metasurface for dynamic terahertz absorption and polarization conversion. The graphene ring presents a certain opening angle (A) along the diagonal of the xoy plane. When A = 0°, the proposed metasurface behaves as a broadband absorber. Numerical results show the feasibility of achieving this polarization-insensitive absorber with nearly 100% absorptance, and the bandwidth of its 90% absorptance is 1.22 THz under normal incidence. Alternatively, when A = 40° after optimization, the proposed metasurface serves as a broadband polarization convertor, resulting in robust broadband polarization conversion ratio (PCR) curves with a bandwidth surpassing 0.5 THz in the reflection spectrum. To tune the PCR response or the broad absorption spectrum of graphene, we change the Fermi energy of graphene dynamically from 0 to 0.9 eV. Furthermore, both the broadband absorption and the linear polarization conversion spectra of the proposed metasurface exhibit insensitivity to the incident angle, allowing large incident angles within 40° under high-performance operating conditions. To demonstrate the physical process, we present the impedance matching theory and measure electric field distributions. This architecture in the THz frequency range has several applications, such as in modulators, sensors, stealth, and optoelectronic switches. THz wave polarization and beam steering also have broad application prospects in the field of intelligent systems.