I heard that some people will use lemon juice to treat pink eye. They said it can help with their symptoms. But I haven't tried this on myself, so I will not suggest you to take lemon juice as the treatment. If you got the contagious pink eye, you should be careful not to infect others for this kind of pink eye is highly contagious. You need to see the doctor timely and apply medicines. Some kinds of pink eye will go away on their own for a few days to few weeks without any special treatment. But you can apply ice on your eyes to calm down the inflammation and lubricate your eyes.
I think you'd better not apply lemon juice to treat your pink eye. Lemon juice has some acid which may hurt our eyes. Maybe you can't bear the pain. There are many eye drops, ointment and pills to treat pink eye. Why not use them to replace lemon juice? But I still think that you should have your eyes checked by an eye doctor and then ask for appropriate treatment.