The phrase "a conversation during which" is correct and usable in written English.
It can be used to describe a specific type of dialogue or discussion that occurs within a defined context or timeframe.
Example: "We had a conversation during which we discussed our future plans and aspirations."
Alternatives: "a dialogue in which" or "a discussion during which".
"The film shouldn't be made," he insisted in a conversation during which he laughed more than spoke from Gotland Island in Sweden, where he was vacationing.
"I hate it," Miuccia Prada once remarked to me about fashion, in a conversation during which we mutually confessed to unease at being compelled by a subject so patently superficial.
Recalling a conversation during which he asked Dietrich which picture of herself she liked best, Mr. Wilson mimicked her cool impatience: "I don't know, I've been photographed so many times".
Mr. Brodsky presented to the jury a conversation during which Mr. Rajaratnam discussed confidential information with another fund manager and then instructed her to "buy and sell, and buy and sell" to create a flurry of trading activity around a particular stock to make it more difficult to detect insider trading.
They struck up a conversation during which Armani promised to become a City fan if Tony bought a counterfeit watch from him.
He was inspired by a conversation during which he asked students if they knew how guns got into their communities, and they nudged in response.
One of my friends from college turned out to work as an editor, so we had a long conversation during which he gave me the ins and outs of the business.
Their meeting, which was to have lasted one hour, grew into a three-and-a-half-hour conversation, during which he nursed one ginger ale and she one glass of white wine.
He added it was important to understand the background, which was that the paper's diary reporter was introduced to Busby as a Daily Mail journalist and had a friendly conversation during which she volunteered information about her forthcoming book and illness and posed for a picture with her husband.
They have a quiet conversation during which John re-lives his life and goes over his relationship with his Mother.
But, according to the ABC, the pair did have a phone conversation, during which Indonesian journalists were "accidentally" allowed to stay in the room.
