Asking very early can make the interaction feel transactional, but it usually does not damage a strong application by itself. The bigger issue is whether the person has enough basis to advocate for you.

If the conversation went well, give the relationship some substance before turning it into a request. You can also ask for advice on the application process first.

A referral is most useful when the person is genuinely comfortable attaching their name to your application.

asking after one chat can feel a bit fast imo.