if you are using the same plugin in the site and are willing to taker dependency on that in your source code, take a look at
EPiServer.Labs.LanguageManager.Business.ILanguageBranchManager.CopyDataFromMasterBranch() method for example (implementation is at
EPiServer.Labs.LanguageManager.Business.LanguageBranchManager). That might give you some options how to do it programmatically.
In the cms (using the Languages add-on) it is possible to create a new language version of a page and duplicate the content from another language. It is possible to do this programmatically?