You may use
c:\>curl -u <user>:<pass> http://localhost:9000/api/settings/values?keys=sonar.branch.longLivedBranches.regex
but note that this is a global setting. If you have different patterns for different projects you may include them in the global regex.
Also note, that the Branch Management was simplified starting from SonarQube 8.1 and the there is no longer distinction between Short-Lived and Long-Lived branches: