NBC is the source for the interview

Short answer: Yes, partially, but it's not quite that simple and it appears to have happened in 1967.

In the show, they show an existing clip from a real NBC interview.

The article does a better job than I, but it essentially boils down to increasing disillusionment (to speak a little too simply) as a result of stagnated progress, the increase of war effort, loss of support from white folk and even some black folk following his objection to the war, and increasing fear of violent outcomes as various people lost patience for non-violence

https://www.nbcnews.com/nightly-news/king-1967-my-dream-has-turned-nightmare-flna8C11013179

I also like to suggest Letter from A Birmingham Jail every time MLK Jr. is mentioned.

Edit:

Here is the report overall:

https://www.youtube.com/watch?v=Ok3BnJeMVwc

Here is MLK's interview: https://youtu.be/Ok3BnJeMVwc?t=1540

Here is the dream commentary: https://youtu.be/Ok3BnJeMVwc?t=1909

Here is the explicit words from his mouth: https://youtu.be/Ok3BnJeMVwc?t=1935

Edit: I made the initial sentence less cautious. NBC is the direct source for this question.