The short answer is yes, under (some interpretations of) panpsychism, rocks and other things might be said to experience a prototypical kind of suffering, which I'd prefer to call "aversive qualia." However, moral duty attaches to sapience (higher cognitive function) rather than merely sentience (subjective experience), so this does not create a moral duty to reduce this sensation for the rock.
First of all, what kind of suffering might a rock experience?
The meaning of a stimulus is given by the effect that stimulus has on the rest of the conscious system. The quale "blue," viewed as a pattern of nerve impulses, has many effects on the brain, causing a change in how it processes certain things. For instance, it might cause the brain to classify an object as "sky." The quale "blue" has the specific subjective nature it has, as a consequence or embodiment of the set of all these possible effects it causes.
It is the same for suffering. Suffering has a certain effect on the brain, and causes a subjective quale because of, and wholly dependent on, that effect.
Now, there are many types of suffering. From a sharp pain in your foot, to heartbreak, to losing a game of chess, to seeing your house burn down, to feeling like there isn't any meaning in your life. It's not just a single sensation, and all these forms of suffering have (somewhat) different effects on the brain.
But there is something that unifies all these types of suffering. They are all things you do not want; things you would take action to avoid, if you could. Suffering is an aversive stimulus.
As I'll explain, there are ways of interacting with a rock that we can interpret as aversive stimuli. However, it is perhaps a stretch to say that any aversive stimulus should be categorized as suffering. Perhaps there are other elements that are important; perhaps we should define suffering as not only an aversive stimulus, but a stimulus that the organism plans ahead to avoid. A rock does not plan ahead, it only reacts.
This is merely a question of terminology. There's no "inherent" definition of the word "suffering" we can refer to. We can call any sensation anything we like.
For now I think I will use the neutral term "aversive qualia" to refer to any sensation produced by an aversive stimulus. This notion captures some, but not all, of the character of human suffering.
To determine whether a certain stimulus to a system is aversive, we could set up hypothetical situations where the stimulus might happen, and ask whether the system moves in a way that reduces or eliminates that stimulus. This is essentially the same as how we could detect when the system is pursuing a goal: aversive qualia are produced by any condition that the system has a goal to eliminate or reduce. Put another way, aversive qualia result from any condition that would go against the goals of the system. The reduction of aversive qualia is an attractor of the system.
A rock has such attractors. For this we should understand that a rock, mechanically, is springy. Like any solid material it has a finite bulk modulus. You can push on it with your finger, and it will push back, like a very stiff spring.
The effect of this springiness is that the rock seeks to maintain its original shape. You push it, it deforms (very) slightly, and the springiness tends to undo this deformation and put the atoms back at the same distances from each other that they were before. Maintaining the rock's shape is an attractor for the rock. From a panpsychist perspective, it wants to maintain its shape, and will respond to a large set of disturbances such that it does so (provided the force does not exceed certain bounds).
So, pushing on the rock, thereby deforming it - which works against this attractor - creates aversive qualia for the rock. It is similar (in a certain limited way, as described earlier) to what we call suffering for humans.
But these aversive qualia are different from human or animal suffering. The rock has no anticipation of the push, no ability to plan ahead and avoid it. It is only reactive. The rock cannot pre-emptively slide out of the way of your finger before you touch it. The rock does not conceptualize the suffering, does not build a model of the world around it to place the suffering in proper context.
An argument from moral philosophy about why it is not as bad to kill animals as it is to kill humans, is that (some) animals do not conceptualize their own deaths. Well, the rock does not conceptualize its own aversive qualia.
So, the moral concept that we should reduce suffering in general - which was derived primarily from the kind of suffering that humans experience, and the complex plans we make to reduce it - is not applicable to the rock.
