It's plated nickel steel like the standard grate is, so no, there's no need to season it.
Either keep it cleaned as best you can, or cook frequently on it and retain a full time layer of grease and oils.
I've also become a fan of scraping it off using a wad of foil held by tongs. Just the big chunks or food bits. I've deposited way too many wire bristles from brushes over the years. Tried the pumice-like bars made for grate cleaning and they do the same thing, not that you asked.
The center section is of course small enough to pop into the dishwasher. (But beware that doing so can really grease up the machine and some nearby dishes.)
I've been using this gadget a lot, specifically to remove the hot center section if necessary or even if it's just too grimy to handle.
http://www.amazon.com/gp/product/B006QY9BJ0/ref=oh_aui_detailpage_o04_s00?ie=UTF8&psc=1