As mentioned, the best option is to replace the chuck. Having different types of chuck is useful regardless.
If that is undesirable for some reason, you could try getting a hex-drive extension. Those generally have round exteriors that will secure in your four-jaw chuck just fine. You may have to cut the hex shaft off of it to fit depending on the extension and the size of your chuck. This will be a little looser than using the proper chuck. Also you probably want to get one of the extensions that has a locking mechanism and not just a magnet, otherwise retracting the hole saw might get a little awkward.
If your hex shafts are not one of the standard sizes, then a socket will often work, either by itself or with a socket extension. But now you're getting really wobbly... Might be best to glue the joints together.
Be really careful with any cobbled-together solutions as a small amount of instability can lead to rapid, unplanned disassembly when rotating at the high speeds generally used for woodworking. Hole-saws shouldn't be too big an issue since they're normally used at lower RPM, but be sure to keep your fingers well back and your face out of the danger plane regardless.