Although this is not usually my field of expertise, Nichelle Nichols (who played Lieutenant Uhura) did have an encounter with Martin Luther King Jr whilst she was filming the series. This is her account of the meeting from an interview:

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Ms. NICHOLS: I went in to tell Gene Roddenberry that I was leaving after the first season, and he was very upset about it. And he said, take the weekend and think about what I am trying to achieve here in this show. You're an integral part and very important to it. And so I said, yes, I would. And that - on Saturday night, I went to an NAACP fundraiser, I believe it was, in Beverly Hills. And one of the promoters came over to me and said, Ms. Nichols, there's someone who would like to meet you. He says he is your greatest fan.

And I'm thinking a Trekker, you know. And I turn, and before I could get up, I looked across the way and there was the face of Dr. Martin Luther King smiling at me and walking toward me. And he started laughing. By the time he reached me, he said, yes, Ms. Nichols, I am your greatest fan. I am that Trekkie.

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Ms. NICHOLS: And I was speechless. He complimented me on the manner in which I'd created the character. I thanked him, and I think I said something like, Dr. King, I wish I could be out there marching with you. He said, no, no, no. No, you don't understand. We don't need you on the - to march. You are marching. You are reflecting what we are fighting for. So, I said to him, thank you so much. And I'm going to miss my co-stars.

And his face got very, very serious. And he said, what are you talking about? And I said, well, I told Gene just yesterday that I'm going to leave the show after the first year because I've been offered - and he stopped me and said: You cannot do that. And I was stunned. He said, don't you understand what this man has achieved? For the first time, we are being seen the world over as we should be seen. He says, do you understand that this is the only show that my wife Coretta and I will allow our little children to stay up and watch. I was speechless.

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So based on this it is quite clear Martin Luther King Jr was a big fan, and saw Lieutenant Uhura as having an important role. I believe that this quote in particular:

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'For the first time, we are being seen the world over as we should be seen.'

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Is quite important as Uhura was not in a subservient position on the crew. Yes, a white man was her commander, but this was in the context of a military rather than social hierarchy. She was an officer, and was an equal to other white lieutenants. Similarly, her rank clearly implied that she had authority over white people who were just ensigns or enlisted, as Starfleet followed a naval structure. Thus African-Americans were considered equal and capable partners in a human endeavor. This was only two years after the 1964 Civil Rights Act was introduced for the purposes of countering segregation.

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Sources

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Star Treks Uhura Reflects on MLK Encounter: https://www.npr.org/2011/01/17/132942461/Star-Treks-Uhura-Reflects-On-MLK-Encounter

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The Civil Rights Act of 1964: A Long Struggle for Freedom: https://www.loc.gov/exhibits/civil-rights-act/civil-rights-act-of-1964.html

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