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.
"There is much beauty that we fail to see through our own eyes teeming with life forms that give us that perception of our reality. Leaves on the trees blowing gently in the wind, or scarily, the waves pounding through high surf, or lightly on a warm summer’s day; that opportunity to sit or swim in the water on a white beach. That comfort to shout, “The universal conscious do you hear me? I am alive, guide me dear logos towards the path of rightnesses.” Earned what has been kept, no longer to be absorbed into a life filled with cold damn winds and that stubborn fog clouding my vision with nothing but darkness."