We theoretically study charge Kondo effects in a quadruple quantum dot. This system has been realized in a carbon nanotube [Nature (London) 535, 395 (2016)] by A. Hamo et al., while it can be also formed in a two-dimensional electron gas (2DEG). The system is in a particular situation where a quadruple dot has twofold degenerate ground states of (n(A) = 1, n(B) = 1, n(C) = 1, n(D) = 0) and (0,0,0,1) charge configurations, where n(lambda) is the electron occupation number of the individual dot lambda = A,B,C,D of the quadruple dot. The two charge states behave as the pseudospin-1/2 states of a Kondo impurity. In the spinless regime, where the real spin of electrons is frozen, for example, by an external magnetic field, the quadruple dot can exhibit a single-channel charge Kondo effect in which coherent charge fluctuations massively occur between the two charge states with the help of electron tunneling between the quadruple dot and its electron reservoirs. The origin of the charge Kondo effect is similar to that of a negative-U Anderson impurity. In the spinful regime, on the other hand, the real spin and charge degrees of freedom couple each other due to interdot electron tunneling between the dots A and B so that the spin singlet is formed in the charge state (1,1,1,0). In this regime, the low-energy Hamiltonian of the quadruple dot system can be mapped onto a two-channel Kondo Hamiltonian having channel anisotropy. In realistic situations of carbon nanotubes or GaAs 2DEGs, the channel anisotropy is so large that the quadruple dot shows a single-channel charge Kondo effect also in the spinful regime at experimentally available temperatures. We compute the temperature dependence of electron transport through the quadruple dot, which is useful for identifying the charge Kondo effects.