Suppose $P_\alpha$ and $P_\delta$ are forcings where $\alpha<\delta$ and these are thought of as members in an iterated forcing. So $P_\delta$ can be thought of as extending $P_\alpha$, i.e. members of $P_\delta$ can be thought of as functions with domain $\delta$. Then if $G_\alpha\subseteq P_\alpha$ we define $P_\delta/G_\alpha$ as the set of all $p\in P_\delta$ such that $p\upharpoonright \alpha \in G_\alpha$.
Suppose that $V[G_\alpha]\models \exists p\in P_\delta/G_\alpha p\Vdash_{P_\delta/G_\alpha} ``\varphi"$
My question is:
Does there exists a $P_\alpha$-name, $\tilde q$ for a member of $ P_\delta $, and a $r\in P_\alpha$ such that (in $V$)
$ r\Vdash_\alpha ``\tilde q\in P_\delta, \tilde q\upharpoonright \alpha\in G_\alpha$ and $ \tilde q\Vdash_\delta \varphi "$?
Yes.
Your assumption $V[G_\alpha]\models \exists p\in P_\delta/G_\alpha\colon (p\Vdash_{P_\delta/G_\alpha}\varphi)$ basically amounts to having a condition in $P_\delta$ which forces $\varphi$. To see this, let $r\in G_\alpha$ and $\tau$ a $P_\alpha$-name, forced by $r$ to be in $P_\delta/G_\alpha$, be such that $r\Vdash_\alpha (\tau\Vdash_{P_\delta/G_\alpha}\varphi)$. Assuming all of the involved posets are separative (which isn't much of an assumption), we must have $r\leq \tau^{G_\alpha}\upharpoonright\alpha$. If we now let $p=(r,\tau^G\upharpoonright(\alpha,\delta))\in P_\delta$, it is clear that $p\Vdash_\delta \varphi$.
So, with this in hand, we can simply take your $\tilde{q}$ to be $\check{p}$ (the check is with respect to $P_\alpha$) and your $r$ to be my $r$.