Some thoughts:
The name Fabricius should be in the dative case.
In English you tell someone, in Latin you tell to someone.
Your translation includes velle, so you are saying more that the king wants to give the thing instead of him going to give it.
In my reading of the original English, the word "would" refers to a future time from the time of saying, not to desire.
Therefore I would get rid of velle and replace it with something of future nature.
While regnum can mean "kingdom", my first impression would be more in the direction of "(royal) authority" or "kingship".
I can't think of a better word, but this nuance of meaning is good to keep in mind in case you might want to dispel misunderstandings of a certain type.
All options I could think of (imperium, sceptrum) suffer from the same problem.
If you want to be careful, a wording in the direction of "a quarter of the land of the kingdom" might be safer.
Se/suus usually refers to the subject of the governing clause (the king) and is to another party (Fabricius).
I would add the dative ei to underline that the gift is going to be made to Fabricius.
Using accusative with infinitive is good here.
Due to point 2 I would shift to a future infinitive.
Based on these, one option would be:
Rex Fabricio dixit se ei quartam partem regni daturum esse.
Whether a future participle should be used depends on the exact interpretation of the English original. The saying happened in the past from now. Does the alleged giving happen at the time of the saying or after it? I took it to happen after, and Sebastian took it to happen at the time. I read "would" to have a future tone like this.
It is common that different languages have different degrees of explicitness. In Latin you have to make a very visible choice between a future and a present infinitive, whereas in English it seems to be more vague. If you are translating a sentence in isolation, you have to choose some things in the Latin that are not determined by the English. If it is not in isolation, then the context will inform the translation.