You can do transparent green.... but not with dye. You need to dye some clear lacquer, and spray it on until you get the shading you want, then use un-dyed clear over that. You can even "burst" the green doing it that way, going from green to greener!
You can get clear lacquer in rattle cans... but to dye it, you'll need a pint or so in a regular can (stewmac.com) and some dye (stewmac.com), and a small spray outfit. If you have an air compressor.... you can get a spray setup at Walmart in the auto-dept. You WILL need clean air - filtered, to shoot lacquer. Dried too. Another option is the CO2 powered "spray units" you can get. You fill the jar with lacquer, and the spray unit screws on top. The sprayer is disposable... but they do a decent job, not too expensive either.
You'll also want a grain filler....to bring out the grain and color it, and to smooth the mahogany. I suggest... filling the grain with a dark filler, several coats, sanded back to wood. Think - 4 to 5 coats. Then clean lacquer over that - prevents the green from seeping into the wood, and also lets you see if you need to correct the "filling" job. Then green, then clear again.