The redox switching of non-alternant azulenequinone/hydroquinone molecules is investigated using density functional theory and the nonequilibrium Green's function. We examined the electronic transport properties of these molecules when subtended between gold electrodes. The results indicated that the reduction of 1,5-azulenequinone and oxidation of 1,7-azulene hydroquinone 2,6-dithiolate lead to a significant enhancement of the current compared to the respective oxidation of 1,5-azulene hydroquinone and reduction of 1,7-azulenequinone, thus “switching on” the transmission. The significance of the position of the functional group on the switching behavior has been analyzed and whether destructive quantum interference exists in the electron transport of the 1,5 position in particular has been addressed. Our work provides theoretical foundations for organic redox switching components in nanoelectronic circuits.