The short answer is no. There are variables involved. If you take a shaft (length unknown), and chuck it in a 4 jaw or good 3 jaw, or a collet, and face it square, chances are pretty good that, provided there is no nib in the center to push it around, a center or spotting drill is rigid enough to get close to center. Then the drill intended to make the hole will generally follow the start given by the center or spotting drill. Exactly on center? No. To get a drilled hole to be exact (relative term in machine work), you should center drill, drill undersized a depth of around 3x the diameter of the finished hole, then set up a boring bar and open the hole to about 0.001 over the desired drill size. Boring the hole will true it up to the center of rotation of the chuck or collet. Then the drill will follow the bored hole closely. So it starts with how accurately you are holding the shaft to begin with. The hole can be true to the spindle axis rotation and still be out or true with the part.
Now consider this: The drill will drift, no matter what you do. Sometimes not a lot, but sometimes quite a bit. So even if you get it to run pretty true at the starting face, chances are more than good that on the exit end (assuming you are drilling the entire length) it will not be on center. If the part you are making requires an accurate ID/OD TIR, then you should start with an over-sized piece of stock, drill it, then finish turn it between centers. If you aren't drilling all the way through, there are other ways to get there, but since you didn't provide all the details, we'll leave that out of the discussion for now.
If the shaft is long, but not so long you can't drill the entire length from one end, yet can't swallow it in the spindle, then you'll need to use a steady rest, and with the same care, get the shaft running true, and face it square and proceed as above.