Here is my current understanding: Q-value is just a value for a particular action taken, and the Value function of a state sums over the Q-value of every action possible in that state.
Would this description be accurate?
First the Q value is a type of value function, it is often called the action value. Both $Q(s,a)$ and the state value function $V(s)$ calculate the expected future return given their parameters, a known environment and a known policy $\pi$ that describes how an agent will select actions in that environment.
In the case of action value $Q(s,a)$, the expected future return is based on the agent taking the action $a$ in state $s$ first, and afterwards following $\pi$. Whilst the state value $V(s)$, the expected future return will depend on what action $\pi$ chooses in state $s$.
There a few ways to write the relationship between $V(s)$ and $Q(s,a)$. If the policy function is deterministic, action in state $s$ given by $\pi(s)$ then:
$$V(s) = Q(s, \pi(s))$$
If the policy function is stochastic, with probability of selecting action $a$ in state $s$ given by $\pi(a|s)$:
$$V(s) = \sum_a \pi(a|s)Q(s, a)$$
This is a sum, as you suggest, but weighted by the probability of taking each action.