RNA-binding proteins are important regulators of RNA biogenesis and function, and recent work shows that chromatin- and RNA-binding proteins can have regulatory roles in transcription. Several approaches have been developed to predict RNA-binding proteins and to identify them experimentally at large scale, including binding to structural elements as G-quadruplexes. Here, we show that the histone lysine demethylase protein KDM5 can bind to RNA through interaction with G-quadruplexes with high affinity despite neither being categorized as an RNA-binding protein nor identified as a G-quadruplex binding protein across numerous experimental large-scale and computational studies. In addition to characterizing the KDM5 G-quadruplex interaction we show that RNA is directly involved in the formation of KDM5-containing protein complexes. Computational predictions and comparison to other ARID domain containing proteins suggest that the ARID domain is directly interacting with both DNA and RNA across several proteins. Our work highlights that non-canonical aspects of RNA-binding proteins are present, and that G-quadruplex RNA-binding could provide a regulatory mechanism for chromatin-binding proteins with impact on transcription through linking protein complexes within the cell.