You are not communicating directly with MySQL, you're communicating with REST server (presumably over HTTPS), and therefore need an SSL certificate, not for MySQL, but for your REST API.
If the REST service is your own, then you need to make it SSL-enabled by installing a certificate on the web server that runs it. If it's run by someone else, then they need to do so.
And if it's your service, and only used by your client, then you can (and probably should if you know how) create your own SSL CA to sign the associated certificate, and then code your application to only trust certificates that you sign. This would mean that you don't need to purchase certificates and also means that you can't fall victim to any sort of SSL MITM attack. Bear in mind that you should only go down this route if you really know what you're doing. Otherwise, just purchase a domain-validated certificate like everyone else does.