We construct one- and two- qubit holonomic quantum gates in an isospectral deformation of Ising-type dimer-chain model. The single qubit is represented by two particles with spin $\frac{1}{2}$; the logical qubit is made of a dimer. We adopt the interaction between those particles as Ising-type interaction, taking a liquid-state NMR quantum computer into consideration. As for the one-qubit gates, we construct the Hadamard gate and gates producing rotations around $x$-axis and $y$-axis. In addition, we construct the controlled-Z gate. Explicitly imposing the condition for the closure of a loop in the parameter space, the set of the attainable rotation angles in a quantum gate is discrete. However, we show such a set is dense and an approximate gate for the desired one can be found with arbitrary accuracy.