If you are talking about an APEX workspace created on http://apex.oracle.com, no, you cannot access the database Oracle hosts from your client machine. You can use the SQL worksheet functionality within the APEX developer environment. If you want to be able to connect to a database using SQL Developer, you'll want to install a local Oracle database and create a workspace there.
As an architectural aside, be aware that APEX users are not Oracle database users and APEX workspaces are not Oracle databases. You can create many different workspaces in a single Oracle database (apex.oracle.com is a single Oracle database hosting thousands of workspaces). And APEX users are not Oracle users so you can't connect to the Oracle database using an APEX user's credentials.