Wikipedia has this to say about self-reference:
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself.
The question has to do with what "object of" means in the following context:
...object of self-reference: 1) a sentence that negates its own truth. 2) an event that negates its own existence...
Since a self-referential sentence refers to itself, the object of the self-reference is the sentence itself. The subject of the self-reference would also be the sentence itself.
To put self-reference in perspective, Wikipedia continues:
Self-reference is studied and has applications in mathematics, philosophy, computer programming, and linguistics. Self-referential statements are sometimes paradoxical, and can also be considered recursive.
It is worth noting that self-referential sentences are only "sometimes paradoxical" and can be useful in recursive algorithms where a subprogram has a line of code that calls itself.
The OP asks:
If it is, can I say that object of self-reference means:
The thing that makes the following self-referential?
For a self-referential sentence the thing that makes it self-referential is the sentence itself.
Wikipedia contributors. (2019, June 9). Self-reference. In Wikipedia, The Free Encyclopedia. Retrieved 13:15, June 30, 2019, from https://en.wikipedia.org/w/index.php?title=Self-reference&oldid=901142171