In a nutshell: No.
This question and others like it (“Can God beat himself up in a fight?”) are clever attempts to prove that the Christian notion of divine omnipotence is inherently contradictory, but on closer examination they prove unpersuasive.
Rephrasing the question is a helpful first step in understanding why we can safely answer it in the negative. As an omnipotent being, God can do anything that is intrinsically possible, a category that certainly includes moving a rock or any other created thing. The attribute of “movable by an omnipotent being” is as inherent in a rock as is “hard” or “composed of minerals.” (This assertion does not rest on Christian doctrine. It is inherent in the definition of omnipotence, regardless of whether or not an omnipotent being even exists.)
There can be no such thing as a rock that is not composed of minerals, because that would be a rock that is not a rock. It is a logical impossibility. Similarly, there can be no such thing as a rock that is immovable by an omnipotent being; even if no omnipotent being actually existed, we could still safely say that if such a being existed, it could move any existent rock.
With this in mind, we can rephrase the question above: “Can God create an object that an omnipotent being could lift [a rock] that is too heavy for an omnipotent being to lift?” Like the inquiry whether God could create a square circle, when properly understood this question devolves into incoherent nonsense, allowing us to answer confidently, “No, God cannot create a rock he cannot lift.” (For more on this topic, see Things God cannot do.)