It is not special phrase as I think, just literature sentence. Haiti is warm green place but very poor and Russia is cold, snowly for full whiteness and many tubercular people live there. This autor thinks such way in this text.

I would say the author is being poetic. green gives us a variety of images of plentiful plants, bushes or trees in a tropical forest or jungle. Such a place may be underdeveloped or undeveloped in a third world country. Green also has overtones of saving the environment. contrasted with tubercular, the word green might imply healthy. Tuberculosis is a lung disease that is highly contagious. It is associated with physical weakness and early death. Is the author saying that it is better to be a poor person in Haiti than to suffer Russian winters? Could be.

Он путешествовал (следовал) вместе с доктором Фармером из зеленой нищеты Гаити в туберкулезную белизну России.