Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact $\tilde O(\sqrt{n})$ time algorithm for LPS, we prove that LCS needs at least time $\tilde Ω(n^{2/3})$ even for 0/1 strings.