Algorithms for online clique detection
First of all I think you mean a maximum clique, not all cliques (and even not maximal cliques). As otherwise e.g in $K_n$ there are $2^n−1$ cliques.

If the question is the one that I said, then there is no polynomial time online algorithm for update (unless P=NP). If there is such an algorithm $A$, given a graph $G$, we can find a maximum clique of $G$ in polynomial time. Just start from a graph on $n$ vertex without any edge and add edges of $G$ one by one and update information in each step by $A$. After adding all edges we have a maximum clique. Note that if we use exponential memory at some steps we can do this, but to write on exponential memory we need exponential time as well.

Maybe there are some heuristics.