"exchanged conversation" is correct and can be used in written English.
This phrase usually refers to a dialogue between two parties, so you could use it to refer to a conversation that two people have together. For example, "The two friends enjoyed an exchanged conversation all night."
Reed and Ingersoll first exchanged conversations in 2005 when Reed found a beat on MySpace that Ingersoll had produced.
Predating Helen Keller by nearly two generations, Bridgman was well known for her ability to exchange conversation with teachers, family, peers, and a curious public.
It's a joyous opportunity to exchange conversation and laughter with people you would only otherwise encounter in the grey heat of the London underground.
As tradition in this region, we all sat on the floor and exchanged general conversation over coffee before the interview began.
Information was exchanged through conversations and stories, which were fast, flexible and suited the rapidly changing world of policymaking.
As they lounge about the place, indulging their macho-geriatric self-pity and lamenting lost youth and missed sexual opportunities, they exchange idle conversation with a fellow guest, conceited young Hollywood star Jimmy Tree Paul Danoo).
One possibility would be to try to find a work-for-keep arrangement so you wouldn't empty the coffers paying for accommodation, for example if you were willing to exchange English conversation for a room.
Gmail calls these exchanges "conversations".
Visiting London to interview for a university post (he's a climatologist), he finds an important-looking file folder left behind in a restaurant by a solitary diner with whom he had exchanged some innocuous conversation.
The number of packets exchanged in the conversation 3.
