Let
- $d\in\mathbb N$
- $\lambda$ denote the Lebesgue measure on $\mathbb R^d$
- $\Omega\subseteq\mathbb R^d$ be open
- $H:=L^2(\Omega,\mathbb R^d)$
- $U$ be a separable $\mathbb R$-Hilbert space
- $Q:U\to H$ be a Hilbert-Schmidt Operator
It's easy to see that for $\lambda$-almost all $x\in\Omega$ $$\tilde Q(x)u:=(Qu)(x)\;\;\;\text{for }u\in U$$ is a Hilbert-Schmidt Operator from $U$ to $\mathbb R^d$, since $$\int_\Omega\left\|\tilde Q(x)\right\|_{\operatorname{HS}(U,\:\mathbb R^d)}^2\;{\rm d}\lambda(x)=\left\|Q\right\|_{\operatorname{HS}(U,\:H)}^2<\infty\;.\tag 1$$
How can we determine the adjoint ${\tilde Q(x)}^\ast$ of $\tilde Q(x)$? In particular, how can we calculate $\tilde Q(x){\tilde Q(x)}^\ast e_i$ for the $i$th standard basis vector $e_i$ of $\mathbb R^d$?
Just curious... Your eq (2) seems to be \begin{align*} \left\langle u,\widetilde{Q}\left(x\right)^{*}y\right\rangle _{U} & =\left\langle \widetilde{Q}\left(x\right)u,y\right\rangle _{\mathbb{R}^{d}}, \end{align*} for $x,y\in\mathbb{R}^{d}$, and $u\in U$.
Fix $x,y\in\mathbb{R}^{d}$, and choose an ONB $\left\{ \varphi_{j}\right\} $ in $U$. (I'm going to assume $\left\langle \cdot,\cdot\right\rangle $ is linear in the second variable.) Then \begin{align*} \widetilde{Q}\left(x\right)^{*}y & =\sum_{j}\left\langle \varphi_{j},\widetilde{Q}\left(x\right)^{*}y\right\rangle _{U}\varphi_{j}=\sum_{j}\left\langle \widetilde{Q}\left(x\right)\varphi_{j},y\right\rangle _{\mathbb{R}^{d}}\varphi_{j}. \end{align*} So this gives \begin{align*} \widetilde{Q}\left(x\right)^{*} & =\sum_{j}\left|\varphi_{j}\left\rangle \right\langle \widetilde{Q}\left(x\right)\varphi_{j}\right|=\sum_{j}\left|\varphi_{j}\left\rangle \right\langle \left(Q\varphi_{j}\right)\left(x\right)\right|. \end{align*}