How to verify integrity of a .git folder?

To verify the object you can use this simply command:

git fsck --full

git-fsck
Verifies the connectivity and validity of the objects in the database

fsck – File System ChecK.
The name is taken from the Unix fsck command

https://git-scm.com/docs/git-fsck

If you want you can also verify the pack separately with but you basically dont need to do it.

git verify-pack [-v|–verbose] [-s|–stat-only] [–] .idx

Reads given idx file for packed Git archive created with the git pack-objects command and verifies idx file and the corresponding pack file.