You will have to be more,specific.
My yard just exited dormancy but I usually don’t apply nitrogen till now, early nitrogen has been shown to cause stem shoot for the whole season.
If you applied something hot like urea or ammonium nitrate, yes, you need to water it in very well, and follow up with a watering in 2 days. These ferts will increase your lawns water usage because they go straight in the root and increases chloroplast and chlorophyll production which requires for carbs which requires more photosythesis, the uptake is very quick and the accelerated plant growth can be seen within 1 day. Urea will linger in the soil for ~3 days and ammonium nitrate ~2 days before leaching through.
If you pt a slow release them the watering requirements are much less. Water after application to wash in the small amounts of urea or an, then business as usual.