Trust is where one person believes in an other person's ability to do the right thing. It can also mean a relationship between one brand or startup to another. It is the one social glue that binds relationships.
Trust is a composition of one's integrity, what one thinks, what one says, and what one does. There is some truth to the saying:
Watch your thoughts, they become words; watch your words, they become actions;
watch your actions, they become habits;
watch your habits, they become character;
watch your character, for it becomes your destiny
The fundamental principle of this ranking order is integrity. Without integrity as the fundamental Jenga building block, the other actions cannot flow from an integral space. Without that integral space, all the rest is just a facade waiting to come crashing down.
It's hard to align all those above things, but the people that I see do the above well are the ones that keep things simple and transparent. They usually say what they mean. They usually are responsive. They usually don't overthink things. Some people call that authenticity, but I think simplicity is the right word. There is also truth to the lyrics of Simple Gifts.
Trust goes both ways in a relationship. You have to give it and have it. I find it easier to give trust first, because that is firmly in my control. Having trust is not within my control, and that takes different amounts of time for different people.
But there are times when you need to prove trust quickly. If there is a sinking ship, a rescuer will demand that a rescuee trust him or her. Of course there is no other option, but that's one way to build trust quickly - by leveraging opportunities that demand trust.
Another way to build trust quickly is for others to say you're trustworthy. This only starts a relationship though and isn't long term sustainable; the above alignment of integrity, thoughts, words, and actions need to be there.