Discoloration of sclera or brown coloration of sclera is a common complaint. It can be genetic depending on the pigmentation as you have mentioned.
The other causes are allergic conjunctivitis
where the conjunctiva
becomes muddy and this can be confused for brown sclera. This can be reduced or treated by treatment.
, pigmentation of conjunctiva are some other causes to be ruled out.
Please do consult your eye doctor for exmination to find out the underlying cause.