Try something like this.
FirstProjectAddIn.ThisAddIn addin = new FirstProjectAddIn.ThisAddIn();
Microsoft.Office.Interop.MSProject.Application prj_app = new Microsoft.Office.Interop.MSProject.ApplicationClass();
if (prj_app != null){
prj_app.Visible = false;
Microsoft.Office.Interop.MSProject.Project p = new Microsoft.Office.Interop.MSProject.Project();
addin.Application_NewProject(p);
prj_app.Quit(Microsoft.Office.Interop.MSProject.PjSaveType.pjSave);
}