You can use
echo "Connected sucessfully to database ".DB::connection()->getDatabaseName().".";
It will give you the database name for the connected database, so you can use it to check if your app is connected to it.
But... Laravel will only connect to database once it needs something from database and, at the time of a connection try, if it finds any errors it will raise a PDOException, so this is what you can do to redirect your user to a friendly page:
Log::error("Error connecting to database: ".$exception->getMessage());
return "Error connecting to database";
Add this to your app/filters.php file.
Hope it helps!!