We propose a new method of the β-γ constraint for quadrupole deformation in antisymmetrized molecular dynamics (AMD) to describe various cluster and shell-model structures in the ground and excited states of light nuclei. To check its applicability, we apply this method to N = 6 isotones, 10Be, 12C, 9Li, and 11B, and find various shell-model and cluster structures as functions of the deformation parameters, β and γ. We also investigate 14C with the β-γ constraint AMD. For positive-parity states, above the ground band, three excited bands with the developed 3α-cluster core structures are suggested. Characteristic structures such as a triaxial structure and a linear-chain structure having 10Be+α correlation appear in the excited bands. [ABSTRACT FROM AUTHOR]