If I move Akali, Deadly Weapon into a battlefield that my opponent controls with a 1 might tentacle at Threshold of the Gray, the tentacle would die from the Akali move trigger but would my opponent or me add 1 energy as per the battlefield?

Neither player adds 1 energy — Threshold of the Gray's trigger never fires.

Here's why: Akali's "When I move" trigger resolves and deals 1 (or 2 if Empowered) to the tentacle before Combat is ever staged. In the following Cleanup, the tentacle is killed by that lethal damage, so when the Cleanup checks for opposing units to stage Combat, only Akali remains — that's a Non-Combat Showdown, and Combat never starts. Threshold of the Gray only triggers "when combat starts here" (Rule 455–456; FAQ #12145).

If the tentacle had survived, then Combat would start and both players — attacker and defender — would each Add 1, not just one of you.