This is a nice exercise. You start by making some reduction steps. Firstly, the BK Selmer group for $T$ is by definition saturated (i.e. it is the preimage in $H^1(K, T)$ of a $\mathbf{Q}_p$-subspace of $H^1_{\mathrm{f}}(K, T \otimes \mathbf{Q}_p)$) so you can invert $p$ everywhere and ask the question for finite-dimensional $\mathbf{Q}_p$-linear representations $V$
Secondly, since you know Shapiro's lemma for the global cohomology, it is sufficient to check that the local conditions match, i.e. that for each prime we have $$H^1_{\mathrm{f}}(\mathbf{Q}_\ell, \operatorname{Ind}_K^{\mathbf{Q}} V) = \bigoplus_{v \mid \ell} H^1_{\mathrm{f}}(K_v, V).$$
The next reduction step is to use Mackey's theorem to write $(\operatorname{Ind}_K^{\mathbf{Q}} V) |_{\mathbf{G}_{\mathbf{Q}_\ell}}$ as a direct sum of pieces indexed by the quotient $G_{\mathbf{Q}_{\ell}} \backslash G_\mathbf{Q} / G_K$. This double quotient turns out to naturally biject with the set of primes $v \mid \ell$, and the corresponding summand is $\operatorname{Ind}_{G_{K_v}}^{G_{\mathbf{Q}_\ell}}\left( V_{G_{K_v}}\right)$. So now we are down to a purely local question: if $M/L$ is a finite extension of $\ell$-adic fields for some $\ell$, we need to check that $$ H^1_{\mathrm{f}}(L, Ind_{G_{M}}^{G_{L}} V) = H^1_{\mathrm{f}}(M, V).$$ This I gather you already know how to do.