Let $G$ be an $n$-vertex graph with $m$ edges. When asked a subset $S$ of vertices, a cut query on $G$ returns the number of edges of $G$ that have exactly one endpoint in $S$. We show that there is a bounded-error quantum algorithm that determines all connected components of $G$ after making $O(\log(n)^6)$ many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least $Ω(n/\log(n))$ many cut queries. We further show that with $O(\log(n)^8)$ many cut queries a quantum algorithm can with high probability output a spanning forest for $G$. En route to proving these results, we design quantum algorithms for learning a graph using cut queries. We show that a quantum algorithm can learn a graph with maximum degree $d$ after $O(d \log(n)^2)$ many cut queries, and can learn a general graph with $O(\sqrt{m} \log(n)^{3/2})$ many cut queries. These two upper bounds are tight up to the poly-logarithmic factors, and compare to $Ω(dn)$ and $Ω(m/\log(n))$ lower bounds on the number of cut queries needed by a randomized algorithm for the same problems, respectively. The key ingredients in our results are the Bernstein-Vazirani algorithm, approximate counting with "OR queries", and learning sparse vectors from inner products as in compressed sensing.