Of course not: DMX and ILDA are very different types of transmission protocols. DMX is very limited in its capabilities, so it's not suitable for any complex signal transmissions.
If you work in ILDA mode, you can only use the ILDA signal. DMX is deactivated at the laser devices as soon as an ILDA signal is available.
Please see also the explanations of the ILDA and DMX signals in the FAQs above.