Is special relativity a special case of general relativity, qualitatively?

Yes, special relativity is a special case of general relativity. General relativity reduces to special relativity, in the special case of a flat spacetime. I.e., general relativity reduces to special relativity, in the special case of gravity being negligible, for example in space far from any objects, or when considering a small enough piece of space in freefall that gravity is unimportant to the problem.

Like special relativity, general relativity also assumes that the speed of light is universal. However, when spacetime is curved, the universality of the speed of light can only be applied locally, within regions of spacetime that are small enough that the effects of gravity aren't important within the region.

Here's a qualitative argument for why special relativity is a special case of general relativity.

When you first learn special relativity it tends to be introduced using the two postulates that Einstein started with. This is a perfectly good basis for SR, but it causes no end of intuitive problems for students. Just search this site for examples. It also conceals the link between SR and GR and IMHO makes it harder to learn GR.

My preferred going in point is that both SR and GR are geometrical theories, that is they describe the geometry of spacetime. SR is one particular geometry while GR allows for many different geometries.

The basis of GR is Einstein's equation:

$$ G_{\mu\nu} = 8\pi T_{\mu\nu} $$

The quantity $T_{\mu\nu}$ decribes the matter/energy distribution, and the quantity $G_{\mu\nu}$ describes the geometry of spacetime. Basically we feed in a matter distribution and solve the equation to calculate a quantity called the metric.

For example if you take $T_{\mu\nu}$ to be a spherically symmetric object like a star then the metric you'll end up with is the Schwarzschild metric that describes black holes. If you take $T_{\mu\nu}$ to be a uniform distribution of matter then you'll get the FLRW metric that describes the universe as a whole. And if you take $T_{\mu\nu}$ to be zero, i.e. no mass present, then you'll end up with the Minkowski metric that describes special relativity (actually there are several metrics that correspond to no matter, but the Minkowski metric is the simplest and best known).

So this is why SR is a special case of GR, because it's one of the solutions to the equations of GR.

You say in a comment above that you're not familiar with the Minkowski metric. All you need to know is that it contains everything you need to know about SR. Time dilation, length contraction, failure of simultaneity, the constant speed of light and lots more can be derived from it. My own personal view is that starting with the Minkowski metric is the best way to understand SR. Just look at all the questions I've answered by invoking the Minkowski metric.