Upgrading 10g database to 11g database

Your problem might very well have to do with incorrect versions or wrong default connection. Make sure that you use the exp utility from the 10g installation and the imp utility from the 11g installation. Also make sure that the environment variables like ORACLE_HOME are correct and that PATH is adjusted for the correct ORACLE_HOME/bin

How does your connect string look like? Is your TNS_ADMIN variable defined? (if not, you might need to have multiple tnsnames.ora files in place, or tnsnames.ora is in a platform specific location (/etc, /var/opt/oracle etc.)) tnsping shows where it gets the sqlnet.ora (next to tnsnames.ora) from.

Even better is to start using expdp and impdp. You can use impdp to directly import your data in the 11g database, using a datasbase link.

As far as I know, Import/Export and Data Pump are command-line-only types of tools. But if you really wanted to use that functionality from within SQL Developer (or SQL Plus), Data Pump uses the built-in DBMS_DATAPUMP and DBMS_METADATA packages. You should be able to call those packages directly, so I would start there. There's documentation available in the Oracle® Database PL/SQL Packages and Types Reference on both of them (DBMS_METADATA and DBMS_DATAPUMP).