We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin.We find the analytical expressions of the first two moments of the walker's position distribution in the long-time limit,which indicates that the variance of the position distribution grows quadratically with walking steps,showing a ballistic spreading typically for quantum walks.Besides,we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge.It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.