Proof of Two Conjectural $q$-Supercongruences Related to Van Hamme's $(\mathrm{G}.2)$ Supercongruence

In this paper, we prove two conjectures concerning parametric $q$- supercongruences related to Van Hamme’s $(\mathrm{G}.2)$ supercongruence. First, we confirm Tang\rq s conjecture \cite[Conjecture $4.1$]{Tang G2} by establishing the corresponding congruences modulo $[m]{\mathrm{\Phi }}_m(q{)}^2$. Second, we prove a conjecture of Wang and Xu \cite[Conjecture $2$]{WangXu24}, extending their $q$-supercongruence modulo ${\mathrm{\Phi }}_m(q{)}^5$ to modulo $[m]{\mathrm{\Phi }}_m(q{)}^4$ when $m\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.