My Druid wants to warn any unsuspecting travelers from taking a shortcut that leads to a hazard that the party is dealing with. Can he use animal messenger to do it?

The animal messenger spell description states:

You specify a location, which you must have visited, and a recipient who matches a general description

Does this language allow my Druid to specify something like, "Go to the crossroads and tell the first person who comes by, 'Avoid the western route'"?

The examples given in the text all imply that the messenger searches for the recipient and immediately gives the message, and the following text implies that there is no waiting around by the messenger.

When the messenger arrives, it delivers your message [...]

The only limitation specifically mentioned regarding time is the 24-hour spell duration.

In your example, yes, but...

The limits on Animal Messenger are the spell duration (24 hours + 48 per slot above 2nd level) and that it affects a Tiny beast. The beast delivers a message of up to 25 words to a recipient matching a "general description" at a location the spellcaster has visited.

Provided the beast can arrive at the crossroads within the duration and a person matching the caster's general description of the recipient appears between the beast's arrival and the end of the spell, the message will be delivered as specified.

"First person who comes by" is a description of the message recipient and therefor allowed. "First person who comes by after 2 o'clock" is not valid as it relies on the beast also marking time rather than simply matching a recipient's description.