Molecular dynamics simulations were used to calculate the equilibrium lattice constant and intrinsic stacking fault energy (γISF) of binary, ternary, and quaternary Ni-based solid solution alloys formed by substituting Ni atoms in the matrix with Cr, Co, and Al. Molecular dynamics simulations were also used to monitor the compression deformation behavior of Ni-based alloys with different γISF values at room temperature. The results show that the relative ability to reduce the γISF is Co < Cr < Al when equal content of Cr, Co, and Al added to the Ni matrix to form binary Ni-based alloys. The γISF decreases with an increase in the alloying element content. The γISF of Ni–Cr–Co alloys is a linear combination of the content of Cr and Co. The addition of Al, complicated the chemical interaction between Ni and the alloying elements, and the γISF of some Ni–Cr–Al, Ni–Co–Al, and Ni–Cr–Co–Al solid solution alloys fluctuate up and down with an increase in alloying element content. The formation of deformation twins resulted from the sequential expansion of stacking fault planes. The process of alloying with elements with a lower γISF means that it is easier to form deformation twins.