Prize collecting Steiner tree on graph without weights on edges
Something seems not quite right.  Your problem can be solved in polynomial time, so I doubt that it is NP-hard.  Let me explain how to solve it in polynomial time.

First note that there exists a positive integer $t$ such that your problem is equivalent to finding $H$ that maximizes

$$f_t(H) = \sum_{v \in V_H} w(v) + t,$$

subject to the additional constraint that $\deg_H(v) \ge t$ for all $v \in V_H$.  Also, $1 \le t \le n$, where $n=|V|$ is the number of vertices in $G$, so there are at most $O(n)$ possible values for $t$.  That means we can try them all and, for each $t$, find the optimal choice of $H$.  Given $t$, maximizing $f_t(H)$ can be done by a simple iterative fixpoint algorithm, where you keep as many vertices of $V$ as possible.

In this way, we obtain the following algorithm:


For $t := 1,2,\dots,n$:

a. Repeatedly delete vertices of degree $\le t$ until there are no more to delete.  In other words, while there exists $x \in V$ such that $\deg(x) \le t$, delete $x$ from $V$.

b. If there exists a single connected component of $G$ that contains all vertices of $Q$, set $H$ to be that component, compute $f(H)$, and if this $f(H)$ is better than anything found before, remember this $H$.


Correctness follows from the above discussion.  This can be implemented so that its running time is $O((n+m) \alpha(n))$, where $n=|V|$ and $m=|E|$ and $\alpha(\cdot)$ is the inverse Ackerman function.