Green is the colour associated with nature, and therefore of God's creation. Many churches at Pentecost will also be decorated with greenery and flowers. God's breath comes to renew all creation. The Holy Spirit is "everywhere present and fills all things..".
Green is also associated with the Holy Spirit, as evidenced in Rublev's icon of the Holy Trinity, where the angel generally recognized as the Holy Spirit (on the right) wears green.
There are other customs that associate the Holy Spirit with nature and the created world. For example, in some Orthodox countries those who have flowers as their name, celebrate their nameday on Pentecost.