9,10-Anthraquinone and 9,10-phenanthrenequinone (PhQ) were grafted onto two kinds of single-walled carbon nanotube (SWCNT) samples having different mean tube diameters by diazo-coupling reactions. The structural details of PhQ-grafted SWCNT (PhQ/SWCNT) samples were analyzed by X-ray diffraction and Raman measurements. It was discussed that a few-nanometer-thick layer of polymerized PhQs covers the outside of SWCNT bundles. The obtained PhQ/SWCNT works very well as lithium-ion battery and sodium-ion battery electrodes, not only at room temperature but also at 0 °C. It should be noted that the cycle performance of the PhQ/SWCNT electrode is much better than that of PhQ encapsulated in SWCNT (PhQ@SWCNT). We also calculated molecular base reaction energies by density functional theory calculations to gain a qualitative insight into the observed discharge potentials of the PhQ/SWCNT electrode.