“Let the universe and all within know that you belong to me, forever.”

The “that” means that the sentence can be split into two phrases by ‘e’, and one part is expressing a proclamation: Let A know that XYZ.

The XYZ part is straightforward - I claim you forever, reH qaDoQ. (DoQ claim (e.g. territory) (v)) I stake my claim on you, forever being the closest possible translation, since tlhIngan Hol doesn’t appear to use belong as in own (property), only ghaj have, possess (v) - and DoQ claim infers an action to turn the object into a possession.

reH qaDoQ means I claim you (singular) forever.

Edit 2020-09-15: It’s been pointed out that the lone protagonist is not aboard the ship accompanied by his parmaqqay, but rather that the ship has claimed the Captain as their parmaqqay. So I’m going with that, and dropping the pronouns.

If the Captain had brought his partner along, and the ship was claiming them both (which would have led to some seriously fun Saturn 3 shenanigans with Klingons), it would have been SaDoQ.

The second half boils down to The universe and everyone knows that. The special ‘e’ pronoun is needed for the sentence as object construction.

luSov ‘u’ Hoch je The universe and everyone knows it

However, since it is a proclamation, the let bit needs the -jaj verb suffix; furthermore, the verb comes at the end of the sentence in proclamations:-

'e’ 'u’ Hoch je luSovjaj Let the universe and everyone know (previous sentence)

Putting them together, you could use

reH qaDoQ 'e’ 'u’ Hoch je luSovjaj

You can get away with dropping the lu- prefix:-

reH qaDoQ 'e’ 'u’ Hoch je Sovjaj

Edit 2020-09-15: It was easier to address queries by just editing the whole article.