This paper presents a scalable IoT architecture based on the edge–fog–cloud paradigm for monitoring the Indoor Environmental Quality (IEQ) parameters in public buildings. Nowadays, IEQ monitoring systems are becoming important for several reasons: (1) to ensure that temperature and humidity conditions are adequate, improving the comfort and productivity of the occupants; (2) to introduce actions to reduce energy consumption, contributing to achieving the Sustainable Development Goals (SDG); and (3) to guarantee the quality of the air—a key concern due to the COVID-19 worldwide pandemic. Two kinds of nodes compose the proposed architecture; these are the so-called: (1) smart IEQ sensor nodes, responsible for acquiring indoor environmental measures locally, and (2) the IEQ concentrators, responsible for collecting the data from smart sensor nodes distributed along the facilities. The IEQ concentrators are also responsible for configuring the acquisition system locally, logging the acquired local data, analyzing the information, and connecting to cloud applications. The presented architecture has been designed using low-cost open-source hardware and software—specifically, single board computers and microcontrollers such as Raspberry Pis and Arduino boards. WiFi and TCP/IP communication technologies were selected, since they are typically available in corporative buildings, benefiting from already available communication infrastructures. The application layer was implemented with MQTT. A prototype was built and deployed at the Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV/EHU), using the existing network infrastructure. This prototype allowed for collecting data within different academic scenarios. Finally, a smart sensor node was designed including low-cost sensors to measure temperature, humidity, eCO2, and VOC.