Let me try to answer in a slightly different way, with a few corrections (unless I'm the one that's wrong). It is a very general statement that, given a compact generator of a stable model or $\infty$-category, the categories of modules (appropriately defined) over the endomorphisms of that generator are equivalent to the original category. This goes back to Rickard's derived Morita equivalence, and the most general formulation is probably that of Schwede and Shipley-- it's also almost surely in Lurie's book.

So, in that sense, $Coh(X)$ is a bit of a red herring -- you don't need to know that your derived category comes from a variety, a scheme or whatever. You just need a compact generator. Once you have that, the question becomes, when do the endomorphisms of the generator look like the path algebra of a quiver. Specializing the the situation of a dg-category, some of the conditions are:

1) The dg-algebra of endorphisms must be quasi-equivalent to an ordinary algebra. In other words, the generator can't have any self-exts outside of degree zero.

2) The identity of the path algebra quiver decomposes into a set of idempotents associated with the nodes of the quiver. Associated to these idempotents are a set of projective modules. The direct sum of these projective modules is the algebra itself. Thus, the generator must decompose into a direct sum. It's the direct sum of these objects that is the generator; each is not a generator by itself.

3) Given two projective modules, $Hom(P_i,P_j)$ is isomorphic to the space of paths between those two nodes (not arrows). This need not be finite-dimensional if you are ok with your quiver having loops. However, a quiver also has a distinguished set of simple representations associated with the nodes that satisfy $Hom(P_i,S_j) = \mathbb{C}^{\delta_{ij}}$. These are just the one-dimensional representations associated with the nodes the quiver.

The dimension of the space of arrows between two nodes of the quiver are given by $Ext^1(S_i,S_j)$. So, if you want a finite number of arrows, that better be finite dimensional. In fact, you can recover the entire quiver algebra by looking at the self-ext dg/$A_\infty$-algebra.

Relating this to the comments, a full strong exceptional collection exactly gives you (1) and (2), but it's not a necessary condition -- you can get quivers for noncompact varieties, for example, a situation that is very common in studying D-branes at singularities in string theory.

One subtlety I've neglected so far is the difference between the unbounded derived category and the bounded derived category. I'm getting more out of my long-atrophied depth here, but I believe the equivalence from the unbounded category descends to the bounded one if perfect complexes are the same as compact objects. This is true, I think, for quasi-compact separated schemes. For stacks, you can see results in Ben-Zvi, Frances and Nadler.