Say “I don’t know, but I’ll find out,” then actually find out. Most of the trust is in the second part.

The second half of that sentence is the half that earns anything. Promising to find out and then not finding out teaches people that your honesty is just a politer way of dropping things, and they quietly stop asking you. Closing the loop is usually small work: one question to the right person, then a message back. Most of the value is in returning before anybody has to chase you.

Write the question down the moment you say it, along with who can answer it. Come back today or tomorrow. If you still cannot find out, say that rather than going quiet.