If a person who knows how to write music imagines a tune in their head, would they be able to write the notes for it? If so, then is it a lot of trial and error to get it exactly right?

Yes.

Music manuscript is very much like verbal script. People fluent in it actually hear the music in their heads the way you hear the words in your head when you read. So if someone fluent in reading music has a melody in their head, yes, the can write it right down.

Bach used to write music at night, silently, when the rest of the household was sleeping. He didn’t need an instrument. Just a quill and ink. Mozart is reputed to have gone to hear a piece sung by a choir that was only performed once every four years, and the music had never been written down. The choir had to learned it by rote. But Wolfie went home after the show and wrote all the parts out from memory. That’s the extreme example, but it makes the point.