It is well known that Vitiligo appears at hair as well as skin in areas affected, but greying of hair is usually a manifestation of the ageing process and results from a progressive reduction in melanocyte function
The association between premature greying and certain organ-specific autoimmune diseases is well documented. The relationship is probably not one of common pathogenesis but on the basis of genetic linkage. It is often stated that premature greying may be an early sign of pernicious anaemia, hyperthyroidism and, less
commonly, hypothyroidism, and all autoimmune diseases that have a genetic predisposition. In a controlled study of the integumentary associations of pernicious
anaemia, 11% had premature greying.
Dawber RPR. Integumentary associations of pernicious anaemia. Br J
As you say, vitiligo sufferers often lose hair pigment (poliosis) in some or all of the patches affected by vitiligo. As generalized vitiligo is most commonly an autoimmune disease, it makes sense to consider that here the hair melanocytes are also being attacked by the immune system, so in this case there is indeed a common pathogenesis. In other cases the hairs in the vitiligo patches may retain their pigment and this is a good sign for predicting response to treatments (as the melanocytes can migrate out from the follicles and repigment the epidermis).
Some kinds of premature greying or whitening can occur in generalized premature ageing (progeria) syndromes like Werner's Syndrome; in this case it is not autoimmune, not patchy, and nothing to do with vitiligo.