At this point in time, the Conversations APIs do not offer the ability to search across all Messages within a Twilio Account or Service.
That said, Conversation Messages are able to be retrieved based on the Conversation and Service that they are a part of.
Conversation Message Resource: https://www.twilio.com/docs/conversations/api/conversation-message-resource#list-all-conversation-messages
Service-scoped Conversation Message Resource: https://www.twilio.com/docs/conversations/api/service-conversation-message-resource#read-all-service-scoped-conversation-message-resources
As the noted alternative would include retrieving Conversation Message details from your own datastore, Conversations Webhooks (in particular onMessageAdded) can be used to reduce the amount of API calls for your account and have message details automatically delivered to your datastore for indexing and searching later.
Conversations Webhooks - onMessageAdded: https://www.twilio.com/docs/conversations/conversations-webhooks#onmessageadded