Hummmmm then we are faced with the legality of using romex to do this.
Many will not really care about that--till after the house burns down, or some other mishap that brings in the investigators. In short, it is illegal to run romex "outside" a wall, though stapling it to structural members such as in a garage is fine. It is illegal to run romex through a wall--from one side to the other. It is illegal to run romex through a doorway or other opening in the wall. Further legalities: It is illegal to use romex as an extension cord--not listed. It is illegal to run an extension cord through a wall, doorway, or other opening in a wall. It is illegal to use an extension cord in continuous use (over three hours.) It is illegal to run any voltage other than low voltage < 40VAC under any of the above conditions. Pick which ever poison suits the situation. I would bring in some UL regulations, however, I seriously doubt the fixture is UL listed, though the components may be--not the same though.
In short, don't bother using romex--it is going to be illegal. Just about any way you try to get the wires out of the closet to the tank is going to be illegal, unless you use conduit, and individual conductors. These laws exist for a reason, generally fire prevention/safety concerns. Violate them if you will.
If you want to extend ballast outputs, use 14 ga, or larger, SOW or SOOW (Service, Oil, Water Resistance or Service, Oil, Other Substances, water resistance) cord, and replace the entire cord from inside the ballast to inside the fixture--do not splice it--and before you do, know where the ignitor is located. At any rate, you will be very hard pressed to legally get anything out of the closet to the tank.
However, and I am going to assume the hobby meanings of HQI, double end pulse start, @ 250watts, and thus an M80 ballast. The max distance, with standard ignitor, to the fixture is 5ft. A long distance ignitor is not available. So unless the ignitor is inside the fixture itself, not the ballast case, you are hosed.
So the short answer to your question is NO it is not allowed, either by the ballast/fixture configuration, or the law.