Problems that provably require quadratic time
Finding an envy-free cake-cutting requires $\Omega(n^2)$ queries. However, this does not directly answer your question as the computational model is different than a Turing machine.

By the way, currently the quickest known algorithm for this problem requires $n^{n^{n^{n^{n^{n}}}}}$ queries, so there is a huge gap from the lower bound - 
probably one of the largest gaps in computer science.