Far as I know it's a reward for the final quest in their quest line, so the answer would be no, you can't.
Console command is the only way. I played nice with the institute on one play through, and you don't get it till the very end. Also, its only for X-01. Very nice though.
Assuming you are using the PC version: set PA_Global_Material_Institute to 1
If you are on the console version, then you are out of luck.
there's an institute logo on it even if u get it , i'm guessing that would bother you abit since u will probably be BoS or Railroad
^This.
It really bugs me that the only way to get this paint is to actually finish the game with the Institute. You get the Minutemen paint for the T-45 as soon as you officially join the Minutemen. You get the Railroad paint for the T-51 as soon as you officially join the Railroad. You get the BOS paint for the T-60 as soon as you visit the Prydwen and officially join the BOS. But with the Institute, you have to commit to them all the way to the end of the game to get their paint job for the X-01. This seems rather...inconsistent.
I used the aforementioned console command to unlock it, otherwise I never would have gotten it.
For what it's worth, the Winterized finish for the X-01 is at least white(ish), but not the pearly white color of the Institute paint job.
Yeah we should get it after finishing the Institutionalized quest to follow how the rest get aquired, kind of svcks in my opinion.
That would be the logical point for it to unlock. I'm not sure why they didn't do it this way.