Why We Would Hunt Them
Humans never tolerate a threat against our most vulnerable population. Throughout human history we have attempted to exterminate many dangerous animals simply because they are dangerous. Wolves, snakes, bears, etc. While these animals are dangerous to hunt, it is far more dangerous to allow them to freely cohabitate with us and risk having to encounter one when you are not prepared to fight it. Or worse yet, risk having one encounter your children as they are playing in the backyard.
Because they are so aggressive and dangerous, the same logic applies to your space whales. Space will be full of defenseless freighters, yachts, space colonies, etc. Many of these may not have FTL drives or military grade sensors at all which means they may not be able to avoid/outrun a space whale. So, part of making a sector of space safe enough for humans to live in will be exterminating anything that is a threat to these unarmed human ships.
This would be a primary objective of any frontier population, for which they would be willing to spend a significant portion of their resources on.
How We Would Hunt Them
Why would some humans want to hunt these creatures even if the possibility of solo killing one is very low and even a fleet of spaceships can easily lose a lot of men during the combat.
Ever heard of an Elephant Gun or maybe a Bear Trap? The difference between humans and space whales is that the threat level of a space whale is fixed, but humans have no problem purpose building weapon systems to the specifications required to kill a specific known threat.
Like the elephant gun, whale hunters could just be massive artillery ships that are way too specialized to be a proper military weapon, but it does one thing very well: it fires a massively powerful shot over long enough of a distance that it can simply snipe the space whales from several light seconds away. Since space whale biology is fixed you just need to make sure that your weapon has a longer range than the whale can jump, and is strong and accurate enough to penetrate its vital organ.
Or you could go the bear trap route. Your hunters may not engage the whales directly but instead send out a bunch traps for the space whales to attack. The traps look like ships to the whales; so, when one comes in to attack it, oops: that "ship" was actually a 50 megaton contact activated nuke. Splat goes the space whale.
So sure, a space whale might be a threat to a fleet of military ships armed with low-power, high precision weapons indented for killing other military ships, but against a weapon system designed around killing space whales, it is hopeless outmatched.
If someone has any idea of what chemical elements can constitute the body of this creature and a plausible explanation of how the only organ works and the possible way humans can use it let me know.
As long as the whale accelerates slowly enough, this organ could be a good old fashioned pocket of lipids and carbohydrates grown through photosynthesis. Since your body can recycle ATP, it just moves faster and faster as it absorbs more energy from the star without actually giving up much mass: just the relativistic stream of particles it shoots out its backside.
The only thing that makes this tricky is that the whale can do FTL jumps. In physics as we know it, there is no source of energy you can self contain that would allow you to meet or beat the speed of light, regardless of how efficient your energy conversion is.
This means you need to break the known laws of physics by either having a self contained "energy" source that is >100% efficient or you need to have some manner of non-Newtonian locomotion that allows >100% efficiency when converting power to work.
Since you described whale FTL as a "jump" I will assume the trick is non-newtonian locomotion. The thing about jump drives is that there is no scientific explanation for how they should work or how much power they consume; so, you could make the power budget for a jump whatever you want it to be. In this case, you could make it so that the biological jump drive that space whales have are some sort of super technology unto itself; so, while humans have to run around using fusion, anti-matter, or even zero-point reactors. It could be that whatever exploit in the laws of physics that the whales use is so efficient that their power source is just good old fashioned solar power stored in fats and sugars. This would explain why the humans can not figure out how to use it for anything useful, because as a power source for human technology, it's really not that good.