Later on, Liam needs Theo again: this time however, he mentions trust to him. "We need to trust you" and Theo's immediate response is that you cannot trust him. That's what he tells Stilinski too: smart people don't make the mistake of trusting him, and trust isn't important right now, because what they need is a temporary ally, not a new pack member.

THEN, months later, Liam mentions trust again, this time to tell Theo that SCOTT will never trust him. "Why do you keep trying to save me?" Technically speaking, Theo has no reason to save him, so Liam finds a reason why. Theo wants a pack, power, whatever. "You think it'll make Scott forget about everything you did? Just let you into the pack? Scott's never gonna trust you."

Now, hear me out. First of all, there's the emphasis on Scott. Why are you saving ME, are you doing it to get SCOTT's forgiveness and trust. Liam's words are basically this: you are helping me because it's useful to you, but it's not. You can't use me to get to Scott again. (Please let us keep in mind that Liam is emotionally distressed right now and didn't get to explode Gabe's skull against a mirror. He's kinda tense and pissed off so it makes sense he'd try to hurt or get a reaction out of someone else because that's litteraly what he does when he's scared)

Second, again, the emphasis on Scott. So this is like, headcanon territory (well all of this is but whatever), but I genuinely believe that at this point, he does trust Theo, and that's why he says that SCOTT is never gonna trust him. And Liam isn't comfortable with trusting Theo: he deflects when Theo points out that he wanted his help, and he makes a point that Theo can never be a member of the pack, he's on the defensive. But he does trust him. He lets his best friend go with Theo in the sewers, he asks for Theo's help as mentioned before. He's aware that he trusts Theo, but also that Theo tried to get him to kill Scott, and that because of this, he should absolutely not be trusting him, or his motives for helping him.

To me the last elevator scene and the events afterwards solidify Liam's trust in Theo: Theo helped, again, when he could be running the other way while everyone takes care of the Anuk-ite, and not only that but he takes Gabe's pain away, an actual act of caring for someone else.