We only have speculation here I'm afraid. Firstly, Martin didn't kill her. To quote from the script:
Martin: I won't shoot her.
Jens: You don't have to murder her. You'll be dismissed. Maybe you be in prison for a few months. Infliction of injury. Abuse. Maybe they'll
conduct a trial. It'll cost you something. But you have to, to save
August. I promise, I'll tell you where he is. You can find out from me
what you want and what I've done but I've always kept my word. I told
you where Mette and the children were. You can still save him.
Following this, Martin shoots her in the upper right of her torso, just below her shoulder. It seems unlikely she would die from this (but again, this is just speculation - the script and show don't confirm it).
Generally, for a criminal case to come to trial the country's prosecution authority will take the case forward. In Sweden, for example, that is the Swedish Prosecution Authority. In Denmark, it is the Danish Prosecution Service. It's conceivable that whichever authority had jurisdiction specifically chose not to proceed with a criminal prosecution against Martin, given the incredible stress and trauma he had been through.
They may have felt there was little chance of a conviction given this trauma, or they may not have had the appetite to pursue the case against a police officer who was so horrifying psychologically affected by the case.
The lady who was shot could probably have some sort of private course of action against Martin, but may have decided not to pursue this if she had been suitably compensated for her troubles.
Of course, all of this answer is pure conjecture. It could be an oversight on the part of the show producers, or it could be that it was not something they felt was relevant to explore after the thrilling drama of the finale.
Either way, "we'll never know" is a fair answer, but I wanted to provide a possible "in-universe" reason.