Getting sick after eating broccoli usually means that you have an allergy to broccoli.
If you always get sick after eating broccoli you are usually allergic to broccoli and are sensitive to salicylates, which is found in broccoli and other vegetables.
You may also be sensitive to even small amounts of salicylates and it's even found in apples.
If eating apples also causes you to get sick then you likely are allergic to the chemical salicylates.
You can get tested for the broccoli allergy by your doctor.
Or it could also be you're just having trouble digesting the broccoli itself as for some people broccoli and other green vegetables can be hard to digest.
Some people just have trouble digesting certain foods and vegetables.