In this paper, a new univariate quasi-interpolation operator is presented by means of construction way with cubic Multiquadric functions. It possesses univariate cubic polynomial reproduction property, quasi convexity-preserving and shapepreserving of order 4 properties, and a higher convergence rate. First, the quasi-interpolation operator ( ) R L x is applied to approximate the derivative of order 1,2,3 m  and its approximation capacity is obtained, i.e., 3 2 0 1 ( ) . m C h h c C c    Second, it is used to construct numerical schemes to solve the diffusion equation. Using the derivative of the quasi-interpolation to approximate the spatial derivative of the differential equation. And applying Crank-Nicolson scheme and back Euler scheme to approximate the temporal derivative of the differential equation. And as 2 ( ) c O h  , the computational accuracy of the scheme is both 2 2 ( ) O t h   and 2 ( ) O t h   respectively. Finally, some numerical examples is given to verify the scheme for the onedimensional diffusion equation. The numerical results show that the numerical solution are very close to the exact solution. Keywords-Multiquadric quasi-interpolation;diffusion equation; shape-preserving property; approximation capacity