The quantization problem for random fractals presents unique challenges due to the lack of uniform geometric scaling inherent in deterministic systems. In this article, we establish the almost sure quantization dimension for a class of $1$-variable (homogeneously) random self-similar measures. Unlike the deterministic setting, where the dimension is derived from a fixed pressure function, we prove that in the random case, the quantization dimension $κ_{r}$ is the unique zero of the expectation of the topological pressure. We rigorously justify this by exploiting the ergodicity of the shift map on the symbolic space to control distortion errors across non-uniform scales. Our results highlight the thermodynamic formalism underlying the quantization of random dynamical systems.