If you work in an organization that's viewed as untrustworthy, you can still earn trust through your own actions and behavior, but it's likely to take a lot longer than if you were in a trusted organization. And, unfortunately, it takes much longer to rebuild a reputation of trustworthiness than it takes to lose it. When people distrust, they tend to adopt a "wait and see" mindset until they're convinced that any improved behavior in those they distrust is permanent and not just a fluke. It is much easier to keep the trust you've earned than to lose it and gain it back.
And that brings me to the factor you can control: yourself. Ultimately, you earn trust by what you say and how you behave—and this is true whether or not others are trusting by nature and whether or not the context is viewed as trustworthy. It's not any one thing that earns the trust of others, but rather a pattern of behavior over time that says, "I'm someone you can trust."
This pattern of behavior entails such things as meeting your commitments (or letting others know in advance if you find you can't), being friendly, communicating coherently, listening with the intent to hear what is being said, being open to other people's views, accepting responsibility for your mistakes, appreciating the efforts of others, treating others with respect, and exhibiting honesty, integrity, kindness, and empathy. And if you want others to trust you, you have to be willing to trust them and to act accordingly.
Basically, earning trust is a matter of doing the things we should be doing anyway, and we can do them without needing anyone's support, approval, or go-ahead. Of course, after earning trust, you can't get lazy; you have to keep on behaving in a way that justifies a continuation of that trust.
One other relevant factor is word of mouth, and this factor can work for you or against you. What people hear about you from the people they trust can dramatically influence how they react toward you. If Tom hears from his trusted buddy Sara that you're great to work with, Tom will be more predisposed to trust you even before he's met you. Conversely, if Sara badmouths you to Tom, you're likely to face a much bigger challenge in gaining Tom's trust. That provides all the more reason to conscientiously and diligently behave in a way that fosters trust.
All this was more than I had time to explain when the woman in my conference session asked me how to quickly build trust, so I simply said to her, "In the brief time that you've seen and heard me during this presentation, do I seem like someone you could trust?" She said yes. I told her, "In that case, contemplate what I've said or done that led you to feel that way, and do those things yourself."
But I wasn't comfortable giving such a glib response. After the session ended, I went looking for her to continue the conversation. I couldn't find her. Maybe she had dashed back to work to quickly build trust.