The localization is a very important task in many internet of things (IoT) based applications. Due to high requirements of communication cost and power consumption, however, existing localization methods are hard to apply for practical IoT applications with low power and narrow frequency. To answer this problem, in this paper, we propose a fully distributed low energy localization method by incorporating the distributed quantization aware least mean square (DQA-LMS) algorithm in a mobile diffusion narrowband IoT (NB-IoT) network. Each mobile network node in this case is assumed to use a low-power and low-resolution analog to digital converter (ADC). First, we address the performance analysis of the mobile networks in the presence of the quantized data. Second, we evaluate the localization performance of the proposed method with differently leveled quantization data and confirm that it can perform suitably even in the one-bit data case. Finally, we collate the localization performance of the Mobile DQA-LMS with the mobile distributed LMS (Mobile DLMS) and show that for all the quantized data cases, the performance of the mobile DQA-LMS is better than the conventional DLMS in terms of mean square deviation (MSD).