The existence of a type of dense subsequence of every dense sequence of eventually zero functions in $\omega^\omega$

Question

Let $G=\{g_n: n \in \omega\}\subseteq \omega^\omega $ where $g_n$ is eventually equal to zero. Clearly $G$ is dense in $\omega^\omega$. Suppose that $Q=\{q_n:n \in \omega\}\subseteq G$ is also dense.

Question: Does there exists a subsequence $Q'=\{q_{n_v}:v \in \omega\}\subseteq Q$ such that $(\forall u\geq n_v)(q_{n_v}(u)=0)$ and $(\forall v \in \omega)(q_{n_{v+1}}\mid n_v=0)$?

Any suggestions?