Autochthonous breeds of livestock are considered a pivotal genetic resource for agriculture, rural development, and food and nutrition security. In the Italian Alps, local livestock breeds are maintained using the traditional alpine farming system based on vertical transhumance, with the use of alpine pastures from late spring to autumn and indoor housing with a hay-based diet for the remaining part of the year. Because of their tight link with the territory of origin, local breeds could be used to biomonitor environmental contaminations. Moreover, animal welfare should also be monitored during transhumance in animals, which are exposed to a sudden farming system change and different types of stressors. For these reasons, this investigation hypothesized that the content of trace elements, heavy metals, and cortisol in the hair of goats changes during vertical transhumance, possibly reflecting different dietary contents and activity of the hypothalamic–pituitary–adrenal (HPA) axis. This study aimed to assess the response of an Italian local goat breed to the change from indoor housing to alpine pasture in summer in terms of hair concentrations of (i) trace elements and heavy metals and (ii) cortisol. The regrown hair of Frisa goats was monthly collected for 2 consecutive years (n = 10 for heavy metals and trace elements and n = 6 for cortisol in 2021, n = 17 for both analyses in 2022), once before vertical transhumance and twice after that event. Hair was then analyzed for trace elements, heavy metals, and cortisol by inductively coupled plasma-optical emission spectrophotometer (ICP-OES) and enzyme immunoassay (EIA), respectively. Data were analyzed by multilevel models. The results showed an increase in As content during alpine pasture (p < 0.01), probably reflecting the soil and water As contents of the grazing area, while Mg, Zn, and Al (p < 0.01) followed the opposite trend, decreasing in the second month after vertical transhumance. Hair cortisol concentrations increased during 2 months of alpine pasture (p < 0.001), indicating an increase in the activation of the HPA axis, in agreement with previous studies. Future investigations can consider a longer study period and the development of ad hoc animal welfare indicators.