You can export your chat/call logs from within the Chat tab:
Open the chat tab, then the conversation you would like to export, then do the following steps:
Android
Tap the 3 Dot icon in the upper right hand corner
Tap Chat Info
Tap the 3 Line icon
Tap Export Chat Records
Enter the email address you want to send the call/chat logs to
iOS
Tap on Profile Name
Tap Chat Info
Tap the 3 Line icon
Tap Export Chat Records
Enter the email address you want to send the call/chat logs to