Now we're getting deep into the kind of stuff I think instructors should go dig into, but probably isn't part of what I'd require for certification. Students don't expect nutrition advice at most MA schools, though I'd expect them to expect that at a school that's helping them prepare for an athletic event (like fight training). Frankly, it's an odd lack of expectation and desire, but there it is.
I'd say it depends on the school. A school that heavily promotes based on fitness? Yeah, clients might come in expecting that. If a good chunk of your ads talk about weight loss you should probably have some nutrition knowledge and a bit of a fitness background.
Instructor certification is a multi component thing though. They need to know what to teach, nutrition might fall under that. As would curriculum, which nutrition might be a part of depending on your schools focus and what you promise clients.
There is also how to teach, which is just as important. That would cover things like learning styles, how to present material, how to drill, how to work with different age groups, class management, discipline, student evaluations & tracking, etc.
As well as intangible things that aren't really "taught" like personality. Probably the most important piece for a instructor IMO. Anyone can be taught how to do and teach a front kick. Not everyone can be taught how to be a instructor that can influence people in a positive way and build strong relationships of trust and mutual respect with students.