"from that quarter"

Never really thought about it. I suppose there are a couple of likely etymologies. For one, the compass is marked by four cardinal points; East, West, North, South, and each segment is termed a quarter, hence the phrase could be analogous to "from that direction". The term "quarter" also is used to designate a geographic or demographic area or subdivision, such as "The Latin Quarter". Another thought is that one's domicile, billet, or assigned place may be referred to as one's "quarters", but all in all I think my first guess is closest to the case.

Yeah. Final answer ... I'll go with "From that direction".

I think the origin of the term is nautical and referred to wind. The direction from which a wind was blowing determined how sails were set relative to the desired course. The term has simple been generalized to imply the origin of any influence, opinion what ever

i got another two sentence from a dictionary:
In quarter, and in terms like bride and groom. --Shak.
and
I knew two that were competitors for the secretary's place, . . . and yet kept good quarter between themselves. --Bacon

here, how to understand " in quarter" and "kept good quarter"
i am confused

Both Shakespeare and Bacon wrote hundreds of years ago. Most of what they wrote is still current usage, but some words and phrases have become obsolete in modern English. I'd say that's the case with the usage of "quarter" here.

In the two examples you quote here, "quarter" seems to refer to the parties being able to get along well with each other -- "share the same space," perhaps.