Is the other website also going to use matukio?
If yes, you can use phpmyadmin (or any other program to connect to your database) and export the #__matukio_xxx tables and the #__compojoom_xxx tables. Import them on the new site and install matukio after that: Matukio should pick up the data. The only problems will be the categories as the categories are stored in the joomla table #__categories. You'll have to recreate the catgories on the new site.