Let be a variable exponent function satisfying the globally log-Hölder continuous condition and L a non-negative self-adjoint operator on L2(Rn) whose heat kernels satisfying the Gaussian upper bound estimates. Let be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels {e−t2L}t∈(0,∞). In this article, the authors first establish the atomic characterization of ; using this, the authors then obtain its non-tangential maximal function characterization which, when p(⋅) is a constant in (0,1], coincides with a recent result by L. Song and L. Yan (2016) and further induces the radial maximal function characterization of under an additional assumption that the heat kernels of L have the Hölder regularity.