While I think its not big deal, I have to disagree. The OP ordered a cheese quesadilla meal from the menu, she should expect to get a cheese quesadilla not a chicken one. Thats like ordering a cheese pizza and it coming out topped with chicken

This has nothing to do with personal responsibilty, this has to do with a simple mistake by the restaurant. I can't actually believe that people would expect the OP to verbally go through a list of ingredients (like meat) not typically found in a cheese quesadilla to make sure it isn't served with it because her dd is a vegetarian. That really has nothing to do with anything, would you expect a non vegetarian that ordered a meatless meal to have to ask for a list of ingredients to make sure there was no meat in the dish? Do the majority of you go into a restaurant expecting everything on the menu may have changed and make sure it hasn't before you order? I know I don't, I read what it says, order and expect that dish to come out.
I don't expect that every ingredient is listed. As I mentioned and a previous poster mentioned, it is pretty common for a grilled cheese sandwich to also include bacon, although it is sometimes not listed on the menu.
I don't think that a patron should ask for all the ingredients of every meal, but again, as others have mentioned, there is a lot that goes into being a vegetarian, and non-vegetarian items are easily overlooked if not asked about.
The OP, since she has very specific, not mainstream needs, has the responsibility to ask one simple question:
"Is this a vegetarian meal?"
That would cover the obvious chicken and the not so obvious things such as lard and other ingredients that others have covered.
I agree the restaurant made a mistake in serving an item with not correctly listed on the menu. However, they replaced it immediately with the correct item. Their responsibility was covered.
The OP, if she is going to be a serious vegetarian in the future, must take the responsibility to question every meal.
Just as if a peanut allergic person might ask if the quesadilla was peanut free. No, peanuts were not listed on the menu, nor would anybody expect peanuts to come out with a cheese quesadilla. But, what about cross contamination with peanut oil or the ppj on the kids menu where the utensils might have been cross contaminated.
If a customer has very specific needs, it is the responsibility of the customer to make those needs known.
The restaurant satisfied their portion of the mistake quite satisfactorily by replacing the offending meal with an appropriate to the customer's needs one. No need to comp the meal.
Since the OP does not seem to even know much about vegetarianism herself, ie: all the other ingredients that need to be monitored for, how can she expect the restaurant to guess her needs?
When she did say that her daughter did not eat chicken (still not mentioning vegetarianism - the kid could just have been a picky eater), the restaurant promptly replaced the item.
Adequate to good customer service in my book.