Currently, I can't find the creating logicApp connector API connection function from Azure SDK document.
If use the Rest API is possible, we could use the following rest API to do that.
put https://management.azure.com/subscriptions/{subscriptionId}/resourceGroups/{resourcegroup}/providers/Microsoft.Web/connections/sql?api-version=2016-06-01
I also test it on my side with postman with following body.
{"properties":{"api":{"id":"/subscriptions/{subscriptionid}/providers/Microsoft.Web/locations/eastus/managedApis/sql"},"parameterValues":{"server":"{yourserverName}.database.windows.net","database":"{databaseName}","username":"{username}","password":"{password}"},"displayName":"{name}"},"location":"{location}"}
How to get the authorization token, please refer to another SO thread.