I don’t think there is any direct way to get the folder's modified data. But you can get the folder related item first, and then get the item’s modified data.
But you still cannot get the folder related item using JavaScript Object Model. You can try to get the items which content type is folder by caml query, then fetch the item whose related file's server relative url is equal to the folder's server relative url, then you can get the detailed information from the item.
var folder = rootweb.GetFolderByServerRelativeUrl("/Shared Documents/test2");
var query = new SP.CamlQuery();
query.set_folderServerRelativeUrl(“/Shared Documents”);
var items = list.getItems(query);
context.load(items, “Include(Title, FileSystemObjectType, File)”);
context.executeQueryAsync(function(){
var itemEnum = items.getEnumerator();
while(itemEnum.moveNext()){
var item = itemEnum.get_current();
if(item.get_fileSystemObjectType()==”1” && item.get_file().get_serverRelativeUrl()==”Shared Documents/test2”){
//do something to get the modified data
}
}
}, function(){})