Hello,

no because the lastmod date is not managed at all by Joomla for menu items.
It's managed for contents and from other third party extensions such as K2, Virtuemart, etc in all cases where there is an information available for the date of last modification we have already thought and provided to include the tag in the sitemap.

If you want to have a lastmod xml tag in the sitemap for menu, you could do using a template override and adding the current time... but it makes no sense.

Cheers!