What is the Q function and what is the V function in reinforcement learning?
$V^\pi(s)$ is the state-value function of MDP (Markov Decision Process). It's the expected return starting from state $s$ following policy $\pi$.
In the expression
$$V^\pi(s) = E_{\pi} \{G_t \vert s_t = s\} $$
$G_t$ is the total DISCOUNTED reward from time step $t$, as opposed to $R_t$ which is an immediate return. Here you are taking the expectation of ALL actions according to the policy $\pi$.
$Q^\pi(s, a)$ is the action-value function. It is the expected return starting from state $s$, following policy $\pi$, taking action $a$. It's focusing on the particular action at the particular state.
$$Q^\pi(s, a) = E_\pi \{G_t | s_t = s, a_t = a\}$$
The relationship between $Q^\pi$ and $V^\pi$ (the value of being in that state) is
$$V^\pi(s) = \sum_{a ∈ A} \pi (a|s) * Q^\pi(s,a)$$
You sum every action-value multiplied by the probability to take that action (the policy $\pi(a|s)$).
If you think of the grid world example, you multiply the probability of (up/down/right/left) with the one step ahead state value of (up/down/right/left).