At the moment Elastic APM .NET agent doesn't automatically create spans for DB operations using ADO.NET (which is what I assume you mean by "using the SqlConnection"). We do have an issue for ADO.NET support in our backlog but there's no planned milestone for it. We do welcome community PRs if you would like to have a stab at it.
BTW which .NET runtime do you use - .NET Framework or .NET Core? The reason I'm asking is because as you can see at the issue I've linked above, an investigation by one of our community members concluded that it might be easier to add support for ADO.NET on .NET Core than on .NET Framework.
Another solution you can try in meantime is to use agent's public API to create spans for your application's DB operations.