We present an investigation of quasar colour-redshift parameter space in order to search for radio-quiet red quasars and to test the ability of a variant of the KX quasar selection method to detect quasars over a full range of colour without bias. This is achieved by combining InfraRed Imager and Spectrograph 2 imaging with the complete Fornax Cluster Spectroscopic Survey to probe parameter space unavailable to other surveys. We construct a new sample of 69 quasars with measured bJ - K colours. We show that the colour distribution of these quasars is significantly different from that of the Large Bright Quasar Survey s quasars at a 99.9 per cent confidence level. We find 11 of our sample of 69 quasars have significantly red colours (bJ - K a?Y 3.5) and from this, we estimate the red quasar fraction of the K a?? 18.4 quasar population to be 31 per cent, and robustly constrain it to be at least 22 per cent. We show that the KX method variant used here is more effective than the UVX selection method, and has less colour bias than optical colour-colour selection methods.