Low-cost biopolymer chitosan has received considerable attention in the field of anion exchange membranes (AEMs) because it can be easily quaternized and avoids the carcinogenic chloromethylation step. Simultaneously increasing the ionic conductivity and improving mechanical properties of quaternized chitosan (QCS) is key for its high-performance application. In this study, new composite AEMs consisting of QCS and functionalized carbon nanotubes (CNTs) were prepared. CNTs were coated with a thick silica layer onto which high-density quaternary ammonium groups were then grafted. The insulator silica coating effectively prohibits electron conduction among nanotubes and the grafted –NR3+ provides new OH− conductive sites. Incorporating 5 wt% functionalized CNTs into the matrix enhanced ionic conductivity to 42.7 mS cm−1 (80 °C) which was approximately 2 times higher than that of pure QCS. The effective dispersion of CNTs and appropriate interfacial bonding between nanofiller and QCS improved the mechanical properties of AEMs, including both the strength and toughness of the composite membranes. An alkaline direct methanol fuel cell equipped with the composite membrane (5% functionalized CNTs loading) produced an maximum power density of 80.8 mW cm−2 (60 °C), which was 57% higher than that of pure QCS (51.5 mW cm−2). This study broadens the application of natural polymers and provides a new way to design and fabricate composite AEMs with both improved mechanical properties and electrochemical performance.