There might be a "temporary glitch" with the LIAT website as flights to other islands in the same time period do, in fact, show up. Try contacting LIAT directly and see what they tell you (Phone: 268-480-5601 or 5602).

If, for some reason they have suspended flights to Nevis in that time period you could take one of their flights to St. Kitts (flights for this routing on those dates do show up on the website) and then take a ferry or water taxi to Nevis.