This has to be one of the funniest questions EVER on MedHelp.. I would imagine the answer to be NO.....
There is no relation of farting and pink eye. Pink eye can be due to conjunctivitis, uveitis, dry eye, blepharitis, angle closure glaucoma, subconjunctival haemorrhage, corneal infection, injury, contact lens wear and allergies. I would suggest you to get it evaluated from an ophthalmologist.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.