I have just discovered an important difference between the FEP-E232 specification and your implementation. Is this intentional? In E232 a link object contains these elements:

"Type": "link","mediaType": "application/ld+json; profile=\"https://www.w3.org/ns/activitystreams\"","href": "https://server.example/objects/123","name": "RE: https://server.example/objects/123"

Your example has these elements:

 "type": "link","mediaType": "application/ld+json; profile=\"https://www.w3.org/ns/activitystreams\"","href": "https://example.com/users/alice/statuses/1","rel": "https://misskey-hub.net/ns#_misskey_quote",

The name from E232 (although marked as "optional") is crucial for a visually nice embedding of the quote in the post. Implementations may replace the value of "name" within the content with the quoted object itself.

I'm not sure about using https://misskey-hub.net/ns#_misskey_quote as a link relation. The original Misskey implementation of quoted posts works differently. It simply uses _misskey_quote as a value in the note to point to the quoted element. (Although off the top of my head I think they now also use quoteUrl, which is used by other implementations.

I suggest concentrating on E232 and omitting this rel parameter. If you want compatibility with other implementations that haven't implemented E232 yet, sending quoteUrl as an additional value should be enough.