if you want to give carrot and are worried about nitrates, you can buy organic ones. I gave carrots to dear daughter when she was about 7-8 mths old.
Fish, you may want to intro later when baby is about 8 mths old cos fish can cause allergies.
Remember to intro only one new food at a time and wait 3-4 days before introducing another new food. This is so that you'll be able to pinpoint any food that causes allergy.