In this paper we write exactly solvable generalizations of one-dimensional quantum XY and Ising-like models by using 2d-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model. © 2022 American Physical Society.