Justin Bieber and Hailey Baldwin took a field trip Thursday to the courthouse where marriage licenses are issued, and Justin had marriage on the brain.

Several eyewitnesses say the 2 were inside the Marriage Bureau in NYC and Justin was emotional. The eyewitnesses say he was crying and at one point said to his fiancee, "I can't wait to marry you, baby."

One person said ... Justin said to a court official, "Thanks for keeping it on the DL." Another claims the court official was a "judge" ... which insinuates they got married, but we can't confirm that.

And yet another person said they heard Justin and Hailey say they were leaving the country.

It's interesting ... because we know they had a clear plan NOT to get married this year.

Young kids in love ... whaddaya gonna do?