I often feel restless in spring, too. It's usually an excited sort of restlessness, and I think it's because I know summer is coming up — a time of year I associate with new things and waking up from a winter slump.
Lack Of Sunlight In Winter
I don't think I suffer from SAD (Seasonal Affective Disorder), but I do tend to find that I have less energy and motivation during the colder, darker months.
If you consider that lack of sunlight is thought to be linked to SAD, then it would make sense that as soon as the days start getting longer, you start to get your energy back and anticipate the lighter days to come. I think this makes me restless because I'm looking forward to having more energy and enthusiasm again!
Even if you don't suffer from SAD, it's still true that vitamin D (the vitamin we get from sunlight) has an effect on our mental health, mood, and energy levels.
In spring, we start to get more sunlight, and so our bodies get more vitamin D. You could be interpreting this springtime energy-boost as restlessness — just enjoy it, and channel that restless feeling into something positive!