I would not call them quartets; they look like doublets of doublets. There is at least one complicating factor here, and there may be two (I am not sure about the second one). This molecule has a para-type (1,4) substitution pattern; therefore, the two hydrogens that are ortho to the methyl group are not magnetically equivalent. Likewise the two hydrogens that are ortho to the carbonyl group are not magnetically equivalent. In order to be magnetically equivalent, the nuclei must have the same chemical shift and be coupled identically to other nuclei. This issue is often ignored in introductory organic textbooks (which is perfectly reasonable). However, nuclei that are not magnetically equivalent do not follow first-order coupling rules.
EDT
I made a mistake in my drawing of this compound when I examined the calculated spectrum. The actual compound is a substituted phenol, but I reversed the position of the carbonyl group and the oxygen atom. When I recalculated the spectrum, it became much more complex.