My saas told me to avoid too much baingan during the first trimester because of 'garmi'. But later, we cooked it occasionally with very little spices. If your family suggests avoiding it, you can stick to other safe veggies to keep everyone happy.
Medically, brinjal is rich in folate and fiber, so it is quite healthy. Some traditional myths say avoid it, but as long as it is clean and thoroughly cooked, it is safe. Always consult your doctor if you have specific dietary doubts.
Beta, brinjal is fine to eat but don't eat it too often as it is believed to increase body heat. Make sure it is cooked well with a little ginger-garlic to balance it out. Try to eat seasonal light gourds like louki instead.
I ate baingan bharta during my second trimester and it was totally fine. Just ensure it is cooked very well so it doesn't cause gas or bloating, which can be very uncomfortable. Take care of your digestion during these months.
Personally, I ate brinjal throughout my pregnancy and had no issues at all. I just made sure it was washed well and cooked properly to avoid any stomach bugs. But do whatever your body feels comfortable with.