Yes. Red cabbage contains a water-soluble anthocyanin that is a pH indicator. At low pH (acid), it's red. It's purple at neutral pH, and goes blue then green as the solution becomes alkaline. You're not going to see the full range here, because we want the drink to be, well, drinkable.
To get the indicator, chop up a cabbage leaf, put it in a bowl with some water, and microwave until it's boiling (or just add boiling water and allow to steep). A purple pigment will stain the water.
Add a teaspoon of cabbage juice -- I mean indicator solution - to the martini glass. Then add very small quantities of baking soda, just enough to turn the solution blue.
Add gin (or vodka) and vermouth (~6:1) to the glass. You should have a pale blue clear liquid.
Give the recipient the drink. Tell them to watch closely, then add a chunk of dry ice. It will sink to the bottom of the drink, and bubble away happily, slowly cooling the drink. It will also neutralise the baking soda and change the colour of the martini.
In the interests of science, I tried some of the baking soda/indicator mix straight. Baking soda tastes salty on its own, and rather overpowers the cabbage juice. The latter by itself doesn't taste of much; a slightly sweet, vegetably taste, nothing like what its violent colour suggests. Overall, it flavours the martini in a way reminiscent of a dirty martini (martini with olive brine added). So you'll only like it if you like dirty martinis, probably a pretty small constituency.