all missions within star clusters are known as exploration mission as they are basically random. you need to have one of the missions selected, ie 'explore the t'ong nebula' as well as the breathing room one.
once you get three of the t'ong and you complete that exploration mission the breathing room one will complete.
dont just do three without taking the nebula specific mission as well.
Sorry, PVE. Exploration in STO is done through the star clusters: Delta Volanis, Arucanis Arm, Hromi Cluster, Khazan Cluster, Eridan Belt, Rolor Nebula, Betreka Nebula, Zenas Expance and Bítran Cluster. They all have "explore three clusters" missions and you just simply fly around the cluster looking for anomalies named for the sector you are in, for instance, in the Hromi Sector, these are called Hromi Cluster Anomaly, and in the Delta Volanis Sector, they would be Delta Volanis Anomaly.
There are two types of anomalies, the first is a cache of sorts and will hold crafting supplies for Memory Alpha content (anomalous readings). The second type of anomaly will come up as (Explore Unknown System).
If you have Explored 3 Unknown Systems then your mission is complete.