To determine which of the given compounds is chiral, we need to understand the concept of chirality. A chiral compound is one that cannot be superimposed on its mirror image, often having a carbon atom bonded to four different substituents.

Let's analyze each option:

1. Compound P: The given structure shows a nitrogen-containing ring with no chiral centers. The substituents are not all different, and thus, it lacks chirality.
2. Compound Q: This structure has a chiral center. The carbon atom is bonded to four different groups: hydrogen (H), methyl (CH₃), ethyl (CH₂CH₃), and a hydrogen atom in a different spatial orientation. This means it is chiral.
3. Compound R: Although it contains a double bond, the two groups attached to each carbon in the double bond are not different. Thus, it is achiral.
4. Compound S: The compound shows two identical methoxy (MeO) groups substituting the positions on the cyclobutane ring, making it achiral as well.

Therefore, among the options, Compound Q is the chiral compound because it contains a carbon atom bonded to four distinct substituents, giving rise to chirality.