When:
During Part 1, on Mêlée Island, right from the start! Be aware of this, or you'll need to play through Part 1 again!
Method:
First thing is first: Pause the game, go into the text / language options, and switch ON the "Writer's Cut" toggle. This makes unlocking the achievement during Part 1 possible.
During any conversation, you MUST make sure to pick the option that states you're looking for the secret of Monkey Island. AS SOON AS YOU SEE THE OPTION, CHOOSE IT! Sometimes you won't get a second chance to select it.
Here's everyone you can talk to. Not all of these will offer the option, and some only after you've mentioned you have a plan / expedition of your own:
The Lookout
Scumm Bar New Pirate Leaders
Scumm Bar Cook
Scumm Bar Pirates
Scumm Bar Door Person
Elaine
Old Pirate Leaders ("Writer's Cut" only)
Voodoo Woman
Wally
Otis
Stan
Locke Smith
Carla
Museum Curator
Scumm Bar Cobb
Scumm Bar Critic
Ghost Pirate Recruiter
Choose the correct option as soon as you see it appear.