Is it possible to have conjunctivitis without the pinkness?

Not likely: Since conjunctivitis means inflammation of the conjunctiva, the thin transparent membrane over the white part of the eye, it would be highly unlikely formit to be inflammwd without the associated dlated blood vessels that cause the "pinkness".

Unlikely: There is a range of light pink to strongly red in the spectrum of conjunctivitis. but the most common forms cause a fairly deep red color similar to the related virus that causes a cold tinting the throat red.