I guess there is no "Android specific" library for backend. If you need to implement simple server then you can look at Ktor for example. But any other web library/framework can be used as well. List of popular libraries is available in
Korlin docs or in
kotlin community awesome list. This library/framework helps you build a server that can handle requests.
When it comes to SQL you have to choose SQL library as well. Usually it is not associated with your web library/framework. But sometimes in very rare cases it is: Spring is a very big complex of frameworks (that it can be called "combine harvester" 😄) that it has its own solutions for interaction with DBs as well. But I would recommend you not to look at Spring without specific need.
A Kotlin community list awesome of DB libraries.
And I also recommend you to ask such questions in
#server. And library/framework specific questions in library/framework specific channels (like
#ktor).