The AAP says you should wait until 8 months to give carrots, but what I've found online says that babies over 6 months can handle nitrates. And I have read that organic produce is slightly lower in nitrates because of the different types of fertilizer that are used in organics, but remember that "organic" can still be a very broad term, one that doesn't always mean "grown without pesticides."
Actually, I recently read that nitrates build up in fresh carrots that are stored a long time, and that this has more to do with the nitrate level than the organic issue does, so it's safer to buy carrots from local growers and picked within the past couple of days--and to use them quickly. I can't, though, remember where I read it, though I've been searching! So it may be more important to buy recently-picked produce (not from the grocery store, where it's shipped from heaven knows where) than to worry about whether it's organic or not. You can wash off a lot of pesticide residue with a sink full of water and a few drops of dish soap, btw. Not that I don't prefer organic when I can get it--at my local farmer's market I have friends who grow and sell produce so I know exactly what will and won't be on/in it. Nice to only have to wash off plain old dirt!