This idiom "she doesn’t suffer fools gladly" means "she can not tolerate fools", however "gladly" gets on my nerves since "she doesn't suffer fools" is very clear to me. It's idiom OK.
but I came across this: "he doesn't suffer fools gladly or glibly",
which I can not quite understand.
Not only gladly is here but yet glibly follows my unwanted word.
1. She doesn't suffer fools gladly
"Gladly" here in effect means "willingly"; it isn't redundant, since she may well suffer fools unwillingly.
The version with "glibly" might suggest that some people deal with foolish behaviour in a verbally dexterous way.
Not a professional ESL teacher.