We consider the stochastic quantization equation associated with the weighted exponential quantum field model (or the Høegh-Krohn model) on the two dimensional torus. Unlike in the case of the usual (unweighted) exponential model, the drift term of the stochastic quantization equation can be both positive and negative, and that makes the equation more difficult to treat. We prove the unique existence of the time-global solution under a certain initial condition by a pathwise PDE argument in the so-called $L^2$-regime. We also see that this solution is properly associated with a Dirichlet form canonically constructed from the weighted exponential quantum field measure.