Apparently people did. From A History of the Horse Drawn Carriage: 'The commonest French coach of this time seems to have been the borbillard...such a carriage was hung low, and would have swung from side to side, giving such passengers as were "bad sailors" a fit of nausea.' (from 1652)

So yes, there was motion sickness in carriages, and people at the time compared it to sea-sickness.