In some practical fields, the usual function values at the interpolated points are not known, whereas the integral values of some successive intervals are given. How to use the integral values of successive intervals to solve function reconstruction is a significant problem. This paper firstly proposes a class of quadratic spline quasi-interpolants from the integral values of successive intervals. It is a direct construction and it is called integro quadratic spline quasi-interpolants. Secondly, we analyze its polynomial reproducing property and give its super convergence property when it approximates the function values at the knots, and so on. Lastly, a type of modified integro quadratic spline quasi-interpolants and its corresponding properties are presented. Experiments show that the proposed quasi-interpolants are simpler and more effective than the existing integro cubic spline quasi-interpolants. Moreover, it can be generalized to integro spline quasi-interpolants with higher degree.