'earnest conversation' is correct and usable in written English.
It can be used to describe a meaningful or serious discussion. For example: The team had an earnest conversation about how to move forward with their project.
Perhaps even an earnest conversation.
When he got there he heard voices in earnest conversation the other side of the door.
As I approached a playing field one recent Saturday I saw a group of boys in earnest conversation.
After about 15 minutes of earnest conversation - and after obtaining Nigel Clough's autograph - my son returned to our table.
At the Olive Garden, she carried on an earnest conversation with eight parents about how to improve the eating habits and fitness of their families.
In October 2007 in Washington DC, a heart expert from Shropshire had an earnest conversation with a reporter from the little-known US medical website Heartwire.
Ornamented with passages from Racine and long, earnest conversations at restaurant tables, Emmanuel Bourdieu's "Poison Friends" is atmospherically and unmistakably French.
What struck me as I listened to the mayors' earnest conversations was how infrequently the public gets to hear the nitty gritty of serious public policy issues.
Danish director Lars von Trier's new film, Nymphomaniac, features a series of earnest conversations about Fibonacci numbers and Bach's polyphonic theory.
People talk extraordinarily quickly and invariably shout; phones are answered every two minutes in the middle of a serious meeting; earnest conversations are interrupted by the arrival of salesmen and even animals.
The football coach, or one of his assistants, will camp on your doorstep, phone you and send you text messages daily, conduct long, earnest conversations with your mom and dad, and arrange for lovely, long-legged coeds to show you around the campus.
