Elements in a less than a value in a subarray Let A be an fixed array of size n. Q(i,j,k) is number of elements from A[i] to A[j] which are less than k. 

Currently I am using segment tree with each node containing sorted array of leaf elements. This answers the query in O((log n)^2) with O(n*log n) space and per-processing.

Is there a way to answer this in O(log n) while keeping the per-processing and space complexity same.
This is essentially the 2D orthogonal range counting problem from computational geometry, which has attracted a lot of attention over the years. Fractional cascading definitely could be applied to reduce the query time to logarithmic while preserving the other big-O bounds. In a nutshell, the idea is to make auxiliary data structures that relate the positions of elements in each parent array to positions in the child array, so that, given the results of a search in the parent, searching the child for the same key is constant-time.