Facebook Inbox does not support showing Quick Reply buttons at the moment, so you might mistakenly think they are not being sent out.
At the moment, Facebook Inbox does not yet support the functionality of showing Quick Reply buttons, which might lead to some confusion.
Even though Facebook Inbox will not show your Quick Replies, you'll notice that these will be visible when the message is viewed in Facebook Messenger itself.
Take a look at the following image from Messenger:
And compare it to the same message viewed in Facebook Inbox:
As you can see, your Quick Replies are actually being sent out. They are simply not visible in Facebook Inbox.