You cannot move SharePoint wiki library from one SPWeb to another even inside the same site collection through web interface. Also you have no opportunity to save wili library as a template with data. The only solution is to move SharePoint wiki library with PowerShell.

There are two PowerShell commands for moving SharePoint wiki library. The first one is to export data and the second one is to import.

To execute commands, you should start SharePoint console with administrator priveledges.

Export-SPWeb -Identity https://spsite/spweb/ -ItemUrl nameofdoclib  -Path C:\Backup\wikidocs.cmp

If this command worked correctly, than execute the second one:

Import-SPWeb -Identity https://spsite/anitherspweb/ -Path C:\Backup\wikidocs.cmp

The result of import will be stored in the log "c:\Backup\wikidocs.cmp.import.log".