I'm playing a UA ranger and have a Giant Centipede as my animal companion. I'm about to increase ability scores for it and need to know if increasing any will effect the DC of the poison.
On page 278 of the DMG, it includes this paragraph:
Alternatively, you can calculate a monster's save DC as follows: 8 + the monster's proficiency bonus + the monster's relevant ability score. You choose the ability that best applies.
For example, if the effect is a poison, the relevant ability is probably the monster's Constitution.
If you take a quick flick through the Monster Manual, you can easily confirm that this formula is used for most monster poisons, including that of the Giant Centipede. (I have yet to find one where it wasn't, but I'll leave it open just in case.)
However, this isn't really your question. The question is, will increasing the ability scores of your companion increase the save DC of its poison?
That is, unfortunately, not really known at this point. The UA article is very unclear on how the increases in both proficiency bonus and ability scores will actually affect the statistics of the animal companion. In particular, it gives no guidance as to whether we are meant to deconstruct the monster and rebuild it with the new numbers, directly add to certain numbers (which numbers?), or some mix of the two.
Even more unfortunately, if we do get clarificiation of these issues, it's unlikely poison will be clarified, because none of the listed options for animal companions have poison.
So, in summary, you're using a UA that isn't fully fleshed out, and going outside the bounds of what it does flesh out. You and your DM are going to have to work this out on your own.