You are dealing with three separate things here:
- Ending a sentence with a preposition (in general):
There are some who claim that this practice is ungrammatical, but this is not true. This "rule" comes from the seventeenth century when grammarians tried to make English grammar more like Latin.
A preposition is a perfectly appropriate kind of word to end a sentence with.
from: M-w Ask the editor video.
- The verb talk has a different meaning with and without a preposition:
You are not the person I want to talk. (1)
Means: I don't want you to talk = be quiet, shut up, don't say anything (although sentence 1 sounds unidiomatic to me).
You are not the person I want to talk with/to. (2)
means: I don't want to talk with/to you, I don't want you as an interlocutor. You may talk all you want, just not with/to me (and even if you do talk to me I don't really feel like responding).
Sentence 2 can be written like this (to keep the same meaning):
You are not the person with whom I want to talk.
- To as an infinitive marker:
A: Do you want to quit? (to quit is an infinitive here).
B: No, I don't want to quit. = No, I don't want to
Here 'quit' is omitted to avoid repetition. The sentence:
*I don't want
is ungrammatical. You don't want what? In your sentence the verb want is transitive, therefore it has to take an object - in this case the infinitive to quit is the object.