In order to efficiently handle the increased luminosity that will be provided by the High-Luminosity LHC (HL-LHC), the ATLAS Muon System was upgraded by replacing its first end-cap station (Small Wheel system) with a New Small Wheel (NSW) detector. The NSW detector provides high-precision muon track reconstruction, as well as information to the ATLAS Level-1 (L1) trigger for data recording. The data collected by the NSW along with other subsystems must be scrutinized to ensure the integrity of the detector, before making it available as "certified data" for “Physics Analyses”. This is achieved through the monitoring of detector-level quantities and reconstructed collision event characteristics at key stages of the data processing chain, using several Data Quality (DQ) tools. This paper, therefore, summarizes the development of the NSW DQ system and presents preliminary DQ monitoring results obtained from the early detector operation during the preparation of the Run3.