The phrase "a special conversation" is correct and usable in written English.
You can use it to refer to a conversation that holds particular significance or importance, often in a personal or emotional context.
Example: "After weeks of tension, we finally had a special conversation that brought us closer together."
Alternatives: "a meaningful discussion" or "an important dialogue".
"This is a special conversation the nation needs to have.
The Columbia University Club of New York and the Columbia Alumni Association (CAA) are hosted a special conversation with Shapiro about his new book in November 2015.
The Columbia Alumni Association and Columbia NorCal, in partnership with Columbia's Science Initiative, invites you to join fellow alumni to a special conversation and cocktail reception with Professor Caleb Scharf, Director of Astrobiology at Columbia University.
Former secretary of state Henry A. Kissinger took part in a "Special Conversation on Sino-America Relations" on Dec. 9 with a group of students from the Jackson Institute for Global Affairs and the Yale Law School.
Former secretary of state Dr. Henry A. Kissinger engaged today in a "Special Conversation on Sino-America Relations" with a group of students from the Jackson Institute for Global Affairs and Yale Law School.
There wasn't a special conversation around that aspect of the character.
Over the weekend, a woman named Caroline from Cleveland, Ohio, tweeted out a very special conversation she had with a guy she met on Tinder.
Members of the Yale community are welcome to join us on Friday, February 2 for a very special conversation about financial literacy with Mellody Hobson, president of Ariel Investments.
Oprah Winfrey and Llewellyn Vaughan-Lee in a very special conversation in Oprah's beautiful oak grove.
So the audience feels by the end that they've had a really special conversation with someone they didn't know".
We had some beers and then a really scintilating and special conversation about all kinds of shit on the walk from the bar to Kellog's Diner.
