$L^3$ Grover search of NTRU variants
Currently there is (at least) one paper in the lattice literature regarding speeding up lattice basis reduction with Grover search: A Faster Lattice Reduction Method Using Quantum Search. Indirectly, there have also been a few theoretical improvements to subroutines of lattice basis reduction, such as speeding up solving exact SVP with quantum search.

In practice though, the current fastest method in high dimensions is probably BKZ, using lattice enumeration with extreme pruning as the exact SVP subroutine. For this combination of methods, I do not believe anyone ever came up with a significant speedup based on Grover searching, as both methods do not involve a straightforward search over a long, predefined list.

Remark 1: Grover searching generally leads to a decrease in the time complexity for solving a hard problem from $T$ (classically) to at least $\sqrt{T}$ (quantumly). With $T$ scaling exponentially in the lattice dimension, this means we need to increase the lattice dimension by at most a factor $2$ to maintain security. Commonly such improvements in the runtime are not considered a "break" of the scheme.

Remark 2: All of the security estimates for cryptography are just that: estimates. There is no proof that no (classical or quantum) algorithm can quickly solve problems which we now presume are hard to solve. We just don't know any fast ways to solve these problems. So we do not know with certainty that cryptography is secure, classically or quantumly.