No, the purpose of intelligence agencies is to gather and analyze information that might be of use to the nations leaders. Spying -- whether on citizens or foreigners, and whether on people located within the country or abroad -- is often a mechanism used to gather information (though collection from public sources is also a common mechanism.)

And there's nothing really special about the foreign vs. domestic (whether by location or citizenship) axis for "intelligence agencies", generally -- while sometimes the agencies that do domestic and foreign intelligence gathering are separate, that's not always the case, and certainly not a defining feature of "intelligence agencies."