White part of my eye turn brown?

Probably little: Unless injured, the white part of the eye (sclera) can develop some brownish tint over time from the proliferation of pigmented cells which are common in the extra scleral layers, particularly in more pigmented people. This is rarely a problem unless there is rapid growth and focal dark pigment which might be a sign of melanoma - very, very rare.

White of the eye: I am not sure why it turned brown. Some people have brown specks in the eye, which is pigment depositions, but without seeing it or getting more history, it is impossible to tell what's going on. Seeing optometrist or ophthalmologist might be helpful.