Frame challenge: You can't have a planet around a green star
There is another question which specifically addresses how you can have a green star. Two of the answers to that question are relevant here.
First is that a 'normal' star will never appear to be green. The reasoning is the same reason that an iron bar heated from a blowtorch will never appear to be green, our eyes are just not that sensitive to green. Wavelengths that activate the green cones in our eyes usually also activate the red or blue ones as well, since green in in the middle of the visible spectrum. Add on to this the fact that Rayleigh scattering in the atmosphere tries to shift sunlight towards blue as it passes through the atmosphere, and the ultimate effect is that a star can't be percieved as green as long as it is also emitting plenty of red or blue light...which all 'normal' stars are.
The second answer does give a plausible mechanism for a green star to develop: that it is surrounded by a cloud of interstellar gas with a high oxygen content. But, this gas cloud must be very large. If the gas cloud was surrounding a star and roughly the size of our solar system, then it would act like...our solar system: it would coalesce through gravitational attraction into planets. If the gas cloud still exits around a star, that implies that planets have not yet formed.
On the other hand, if the gas cloud was enormous like a nebula, it could remain stable for some time, and perhaps catch a rogue planet that was shot into interstellar space by some long ago migration of giant planets. Unfortunately, now we run into the problem that this gas cloud does not emit very much light. The who principle behind the green light is that the cloud absorbs most of the radiant energy from its interior, and only emits the light in the green band, which is why it appears green.
The gas cloud in the linked question, NGC 6826 has an apparent visual magnitude of 8.8 at a distance of 2000 ly. This translates to an absolute magnitude of -0.1, about 100 times as bright as our sun. But, with the inverse square law for luminosity, a planet would need to be about 10 AU from this object to receive the same light energy as our planet (at 1 AU) does from our sun. However, NGC 6826 has a radius of about 0.2 light years, around 13,000 AU. Thus, a planet getting the required light radiation for photosynthesis would be well inside the gas cloud. In short order, the gas cloud with leach the planet's orbital energy through collisions and that planet will end up approaching or crashing into the sun.
Therefore, I conclude that there is no reasonable mechanism to have a planet circling a green star, so the answer to the headline question is: No, you can't have purple plants (or any plants) on a planet orbiting a green star (because that planet does not exist).