As a Green Witch myself, I usually prefer to use my hands when I perform green magick. Touching and handling the ingredients helps me to feel grounded, and I visualize holding the items in my hand (especially plant materials like herbs/spices, crystals) as connecting me to nature 
However, I imagine that a staff (or wand or athame) can be incorporated to help channel, direct, and enhance any green magick spell or ritual you’d like to cast. It is easy to see how a staff of wood can connect with the nature of green magick 
Different types of wood have different magickal properties that could enhance your spellcasting. When choosing wood for your staff, you might consider Rowan, Birch, or other Sacred Irish Trees 