The effects of poloidally inhomogeneous heat sources are investigated through a gyrokinetic formula in collisionless toroidal plasmas. A gyrokinetic dispersion relation is newly derived under the assumption that equilibrium parallel heat flows are generated to remove the injected poloidally nonuniform heat source. The dispersion relation is numerically solved, considering both inboard and outboard heat source injections. In the case of the inboard source injection, both Stringer spin-up and geodesic acoustic mode (GAM) are excited. Conversely, outboard injection leads to the emergence of a heat source-driven GAM (referred to as Q-GAM), featuring a frequency around half that of the standard GAM. Various physical quantities of the Q-GAM, such as mode frequency and source threshold, are analyzed through parametric scans. The Q-GAM exhibits similarities with the energetic-particle-driven GAM (EGAM), particularly in its frequency range, and both belong to one of the strong Landau damped poles. Despite having distinct driving mechanisms and structural differences in parallel velocity and poloidal coordinates, the response function of the perturbed parallel pressure to the potential, mainly contributing to the destabilization of each mode around half of the GAM frequency, is derived to have a similar form for both the Q-GAM and EGAM cases.