IIRC you can do a SVG output from QGIS (qgis.org) and open that in Illustrator. It would just be the geometry data though, no attributes, which is one of the reasons why I love MAPublisher so much
The layout composer feature in Quantum Geographic Information Systems remains at this point less then the desired output that a cartographer wants to see before exportation into Adobe Illustrator. It's understandable we are talking about about a free product. Yet "I" feel that despite the major advancement of the software near to the legacy & out of its league Manifold System that there is a disappointing aspect to some of the functionality.
It is correct that MAPublisher tends to work such a way to produce faster results, readable geometric files, database input, yet a starting cartographer should not pursue this avenue beyond the obvious costly expense for the software & yearly maintenance. The basic principle must be explored & learned by the learner; subjects such as digitizing, corrective text placement, trivium rules in relation to cartographic principles, designing, & other techniques beyond my conscience for the moment.
No matter the selection of choice by this individual we all can take joy in the thrill of the creation & the blending of the feminine & masculine brain.