Note:

You need Digital Engagement Messaging to try the below. To know more about it check – https://www.infallibletechie.com/2020/04/digital-engagement-messaging-in.html

Agent-initiated outbound messaging is available only on the SMS text message channel.

1.Make sure the Start Conversation Quick Action is added to the page layout.

2. Make sure the record from where you are trying to click Start Conversation(Contact or Lead) is linked to the Messaging User record.

3. Make sure the Messaging User Consent Status is not Opted Out.

4. Make sure the Agent who is trying to start the conversation is part of the Queue to which the Messaging Channel is configured.

Check this for more information – https://help.salesforce.com/articleView?id=messaging_agent_initiated_outbound.htm&type=5