What is a permission? The clue is in the name. It's a permission from the rights holder of a work -- whether that be the author, the publisher or a literary agent -- to take an extract (or asset) from one work and use it in another without infringing copyright. It is sometimes referred to as a limited use license, as there will be restrictions on the number of times and contexts the quote can be used in.
If you are a rights holder, people can and have to contact you for permission to use extracts from your work. It is up to you to set the parameters of the context this can be used in (price, duration and others), and you are completely at liberty to not allow someone to use a permission. Likewise, if you seek to get permission to reuse a quote, you have to get in touch with the rights holder.