The $A$-polynomial encodes hyperbolic geometric information on knots and their complements. Cooper, Culler, Gillet, Long, and Shalen defined this polynomial using representation theory, then Champanerkar introduced a geometric way to compute it. His method was based on an ideal triangulation of a knot complement and the Neumann-Zagier matrix. Later, Dimofte outlined a process of symplectic reduction to compute the $A$-polynomial and then quantized it. For the classical version, Howie, Mathews, and Purcell discovered a way to convert Champanerkar's equations into simpler Ptolemy-like equations that can be read off almost entirely from the Neumann-Zagier matrix. However, the quantum $A$-polynomial remains mysterious and little is known about whether it carries extra geometric information from its classical counterpart. In this talk, we introduce new tools to bridge this gap by presenting a quantized upgrade of these Ptolemy equations. Furthermore, we will discuss their connection to skein theory by utilizing the Heegaard surface of a knot complement constructed from a given ideal triangulation.