As referenced from Wikipedia:
The Indigo Tribe is one of the nine Corps of the emotional spectrum
within the DC Universe. Using indigo power rings and power staves,
they are able to wield the indigo light of compassion. They've been
depicted as being major participants within the Blackest Night
crossover event; however, their reclusive nature makes them the most
elusive group among their peers (comparatively limiting their
depictions within DC titles).
The Indigo Tribes encounter finds the Green Lantern of Sector 3544 grievously and perhaps mortally injured. It can be assumed that like any other color-wielder, The Indigo Triber should have been able to tell if the Green Lantern was mortally wounded. If he was already spitting up blood, had open head injuries and barely capable of speaking effectively, the Indigo Tribe agent may have decided the most compassionate thing to do for that Lantern was to end his suffering. Since neither Lantern appeared to be much of a threat to them, it was certainly not fear of the Lanterns that prompted her behavior.
I am certain that the slaying of the Green Lantern would not have been a good thing, especially in light of the circumstances, however, in their mind, ending the suffering was for the greater good, as they saw it.