No. Every time Jupiter rotates everything on the planet moves all the way around it. Half the time any given location is on the Sunward (day) side, and the other half it is on the other (night) side. And even though, like cloud systems on the Earth, the position of the Red Spot gradually changes, its movement isn't fast enough to significantly affect that statement.

So, since the rotation period of Jupiter is just under 10 hours, the Red Spot is somewhere on the day side for a little less than 5 hours, then on the night side for a little less than 5 hours, over and over, for however long the Spot still exists.