TL/DR - some hash functions are insecure against a quantum computer, but many other commonly used hash functions are likely not broken with a quantum computer.
Indeed there are some cryptographic hash functions based on number-theoretical problems, such as $x^2\pmod N$, that can be exponentially broken with a quantum computer. Here, $N$ is the product of two large prime numbers $N=p_1\times p_2$, and $x$ is a binary representation of the string to be hashed. Knowledge of the secret pair $(p_1,p_2)$ is enough to invert the hash; a quantum computer could run Shor's algorithm to factor $N$ and hence learn the pair $(p_1,p_2)$.
But, as explained by @Callum, many commonly used cryptographic hash functions such as
SHA256 are only known to be susceptible to brute-force style attacks. A quantum computer does give a speedup against such brute-force attacks, but "only" a square-root, Grover style speedup.
It's an interesting problem to know precisely how much structure is needed to get an exponential speedup to crack hash functions.