#Fedilab supports the quoting mechanism for long with some softwares that introduced it.
This option is currently hidden in "Extra features" from Settings.
We opened it to #Mastodon in timelines (ie: messages containing quotes), but it is not yet possible to post quoted messages with Mastodon.
Also Mastodon seems to have introduced a degree of privacy. The API returns if the OP accepted to be quoted.
Fedilab already supports this privacy parameter by displaying only accepted quotes.