Hi Matt Matt, the Dandenong Creek bike trail goes directly under Greens Rd. but I wouldn't recommend it for regular road bike use as it is gravel, and regularly gets erosion cracks that can swallow narrow tyres. Some alternates would be Frankston-Cranbourne Rd, to Westernport Hwy, to South Gippsland Hwy, to Greens Rd. This has a bike lane or wide sealed verge for the majority of its length, although like Frankston-Danenong the South Gippsland can get a bit feral during peak hours, not too bad once it jams up though.
Also their is the Nepean Hwy/Station St. route to Edithvale Rd. which becomes Springvale Rd. Turning right at Governor Rd/Braeside Rd. will take you into the western end of Greens Rd. hope this helps.