The lodgix plugin supports localization for the translation of plugin specific output. The WPML plugin will need to be setup and configured within Wordpress.
Property Specific Translations
Within Lodgix, under Settings > Important Settings, at the bottom, there is an option to select which languages should be supported..
Once selected, translations may be entered within the setup area of each property by clicking on that nation's flag in the upper right corner of the screen.
Advanced - Changing or Adding Translations
There are fixed items in the plugin such as menus, calendars, amenities, etc.. which Lodgix offers translations within the plugin. Additional language translations can be changed or added using ".po" and ".mo" files.
To alter (or add) a translation simply copy the ".po" files from "<lodgix plugin folder>/languages" to "wp-content/languages" and alter the file using POEdit or another po editor and then compile the ".mo" files.
Note: poedit is located here http://www.poedit.net/. Open the .po file and when saved, the .mo file is generated. There are numerous sources of tutorials online, as most applications, desktop or online use .po, .pot and .mo files for localization. Here's a tutorial from drupal: https://drupal.org/node/314231
For German for example, just copy p_lodgix-de_DE.po, edit it, compile it and it's ready.
Using this method you can alter the plugin to display other languages. Simply create a PO file for that and fixed strings on the plugin will be translated (menus, etc).