It really depends. Technically marchrophages can break it up. But it could turn into pnuemonia if the piece is too large. Pneumonia could theoretically cause fibrosis. Depending on the size if I were you I would get a chest x-ray, if it's anything large enough to cause issues, it will be visible. If not since it's an organic substance it should be broken down just fine.
Thank you for your answer. You comment made me rest. I was so worried since I inhaled also tiny peace of carrot. (Thanx to DrakeLucian for the question)
lol, I'm sorry for giggling. If a substance manages to get passed your major airways without causing a blockage, it will soon be removed by the body. You have a natural layer of mucus continuously moving over the surface of your lungs, removing dust etc. Larger problems are broken down by white blood cells that mutate into large macrophages. These literally gobble up rubbish and then die to be removed by the mucus. So, your carrot will be long gone. If you imagine how much debris we inhale in our lives, dust, smoke and other substances, we would all need our lungs sucked out at least once a week.