This paper investigates the thermodynamical topology of the quantum BTZ (qBTZ) black hole using the concept of generalized free energy. The study introduces two topological numbers to characterize the thermodynamics of the qBTZ black hole. The first topological number, denoted by z, is derived from the free energy and provides some local physical insights, but lacks global significance. The second topological number, based on the entropy expression of the generalized free energy, offers a more meaningful interpretation of the physical implications. This result highlights the natural choice of entropy as the domain variable for the generalized free energy. The analysis reveals a topological transition associated with the thermodynamical stability of the "cold" black hole state of the qBTZ black hole. Furthermore, the study suggests the existence of topological numbers beyond the conventional values of ±1, 0. The thermodynamical topology of the qBTZ black hole is classified into three categories associated with topological numbers: +1, 0, and -1. These numbers correspond to the difference between the numbers of stable and unstable phases exhibited by the black holes. The qBTZ black hole is considered a thermodynamical system, inheriting certain thermodynamical properties from its bulk counterparts. Unlike the Hawking-Page phase transition observed in the BTZ black hole, the qBTZ black holes demonstrate a more interesting phase transition, the reentrant phase transition. As the temperature increases from zero, the system undergoes a transition from a thermal AdS phase to a black hole phase, and eventually returns back to the thermal AdS phase. This fascinating behavior is a distinctive feature of the qBTZ black holes. The study reveals several remarkable characteristics of the qBTZ black hole thermodynamics that distinguish it from the classical BTZ black hole. Unlike its classical counterpart, the entropy of the qBTZ black holes is finite, and does not exhibit a monotonic behavior with respect to the black hole parameter z. This observation has prompted us to re-examine the thermodynamical topology, leading us to identify a topological transition associated with the qBTZ black holes. Furthermore, through the study of qBTZ black holes, we have gained a deeper understanding of the significance of topology in the context of black hole thermodynamics. This investigation allows us to explore the relationship between the thermodynamical properties of black holes and their underlying topological features, shedding light on the implications of topology in this fascinating field.This paper investigates the thermodynamical topology of the quantum BTZ (qBTZ) black hole using the concept of generalized free energy. The study introduces two topological numbers to characterize the thermodynamics of the qBTZ black hole. The first topological number, denoted by z, is derived from the free energy and provides some local physical insights, but lacks global significance. The second topological number, based on the entropy expression of the generalized free energy, offers a more meaningful interpretation of the physical implications. This result highlights the natural choice of entropy as the domain variable for the generalized free energy. The analysis reveals a topological transition associated with the thermodynamical stability of the "cold" black hole state of the qBTZ black hole. Furthermore, the study suggests the existence of topological numbers beyond the conventional values of ±1, 0. The thermodynamical topology of the qBTZ black hole is classified into three categories associated with topological numbers: +1, 0, and -1. These numbers correspond to the difference between the numbers of stable and unstable phases exhibited by the black holes. The qBTZ black hole is considered a thermodynamical system, inheriting certain thermodynamical properties from its bulk counterparts. Unlike the Hawking-Page phase transition observed in the BTZ black hole, the qBTZ black holes demonstrate a more interesting phase transition, the reentrant phase transition. As the temperature increases from zero, the system undergoes a transition from a thermal AdS phase to a black hole phase, and eventually returns back to the thermal AdS phase. This fascinating behavior is a distinctive feature of the qBTZ black holes. The study reveals several remarkable characteristics of the qBTZ black hole thermodynamics that distinguish it from the classical BTZ black hole. Unlike its classical counterpart, the entropy of the qBTZ black holes is finite, and does not exhibit a monotonic behavior with respect to the black hole parameter z. This observation has prompted us to re-examine the thermodynamical topology, leading us to identify a topological transition associated with the qBTZ black holes. Furthermore, through the study of qBTZ black holes, we have gained a deeper understanding of the significance of topology in the context of black hole thermodynamics. This investigation allows us to explore the relationship between the thermodynamical properties of black holes and their underlying topological features, shedding light on the implications of topology in this fascinating field.