In this study, we are concerned with a method for constructing quantum-based adaptive neuro-fuzzy networks (QANFNs) with a Takagi-Sugeno-Kang (TSK) fuzzy type based on the fuzzy granulation from a given input-output data set. For this purpose, we developed a systematic approach in producing automatic fuzzy rules based on fuzzy subtractive quantum clustering. This clustering technique is not only an extension of ideas inherent to scale-space and support-vector clustering but also represents an effective prototype that exhibits certain characteristics of the target system to be modeled from the fuzzy subtractive method. Furthermore, we developed linear-regression QANFN (LR-QANFN) as an incremental model to deal with localized nonlinearities of the system, so that all modeling discrepancies can be compensated. After adopting the construction of the linear regression as the first global model, we refined it through a series of local fuzzy if-then rules in order to capture the remaining localized characteristics. The experimental results revealed that the proposed QANFN and LR-QANFN yielded a better performance in comparison with radial basis function networks and the linguistic model obtained in previous literature for an automobile mile-per-gallon prediction, Boston Housing data, and a coagulant dosing process in a water purification plant.