I cut down a red cedar tree and a pine tree. Will it hurt my garden vegetables if I mulch up the greenery before it dies and then spread it around my vergetables? I was not sure if there is a difference between mulching dead vegetation versus live vegetation.

I think this is the best kind of mulch.
I would advise you that if the pieces are big they may become a nuisance and they’ll take a long time to break down. You might consider putting it down in the walkways and using alfalfa straw around the vegetables.
Tony M