We propose a scheme to implement a two-qubit Grover quantum search algorithm.The novelty in the proposal is that the motional state is introduced into the computation and the inteal state within a single cold trapped ion.The motional and inteal states of the ion are manipulated as two qubits by the laser pulses to accomplish an example of a Grover algorithm based on the two qubits.The composite laser pulses that are applied to implement the Grover algorithm have been designed in detail.The issues conceing measurement and decoherence are discussed.