Can I mix Fantasy and 40k in a game?
Unfortunately they are completely different rulesets.  While they have a few similar mechanics(rolling to hit/wound or making characteristic tests), the structure of the games do not synch up.  Turn structure is different. Ranged weapons, saves, movement, and combat are all different.  Applying some fairly common rules in one system to the other can have game breaking results.

Even if you did build a bridge between the two systems the armies would just not play well against each other.  Stats aren't balanced between the two systems.  They have different baselines(Armor saves in fantasy are mostly 6+, if you have one at all, and in 40k 2+ and 3+ are not uncommon). 

The closest you could get would be to have one person play Chaos Demons.  The same demon models can be used in fantasy and 40k with the appropriate army book.  It's a little awkward because 40k and Fantasy use round and square bases respectively, but most players would give you a pass in friendly games.