I encountered Ms. Zeng in the afternoon, s. She was not only one of my English teacher in middle school but also a close friend and colleague of my father.

I less formal way of saying "I encountered Ms. Zeng..." would be "I ran into Ms. Zheng..."

She had heard about my quitmendropping out, and also asked me why I desirewould want to study abroad when the pandemic is so severe outside China.

Not sure what you mean by "quitment", so I replaced it with "dropping out", which would refer to quitting something, for example, a course of study.

She said as she always didoes, "Try to find a job and insist for years, then you will get a good improvementstick with it, then you will improve your situation.

Using "did" vs. "does" here implies something about your past and present history with Ms. Zheng. Using "did" implies she is someone who in the past gave you advice, but hasn't done so currently. Using "does" implies she currently occasionally gives you advice.

After all, women need to get married and gihave birthchildren.

The phrase "give birth" is technically correct, but people would say "have children" or "have a child" instead.

If you insist ion going abroad, you will beget too old to complete thesedo these things."

The preposition used with "insist" is "on", not "in".

She thinks I am stubborn, s. So do I.

This needs to be two sentences, unless you put a conjunction between, e.g., "She thinks I am stubborn, and so do I." But I think using two sentences is stronger.

Persuadesion

There is nothing wrong with the heading you gave, but in English, generally we prefer to use a noun instead of a verb for the heading.

I encountered Ms. Zeng in the afternoon, she was not only one of my English teachers in the middle school but also a close friend and colleague of my father.

She had heard about my quitmentresignation, and also asked me why I desired to study abroad when the pandemic is so severe outside China.

There's no such word as 'quitment'.

She said as she always did, "Try to find a job and inpersist for years, then you will get a goodr conditions will improvement."

If you insist in going abroad, you will be too old to completefulfill these."

Persuade

Persuade on it's own doesn't mean anything.
'Persuasion' might fit better.

I encountedbumped into Ms. Zeng in the afternoon, she was not only one of my English teacher in middle school, but also a close friend and colleague of my father.

'bumped into' if you meet someone by chance, in the street or the cafe.

She had heard about my quitmentresignation, and also asked me why I desirewant to study abroad when the pandemic is so severe outside of China.

quitment isn't a word. Resignation?
'She had heard about me quitting my job,'

She said as she always did, "Try to find a job and insisstick at it for years, then you will get a goodsee great improvement.

'She said as she always did' works but is a bit strange.
'She gave her usual advice'
'She repeated her mantra' works here.

After all, women need to get married and gihave birthchildren.

It's not true that women need to get married and give birth in this day and age, you could write:
'After all, women are expected to get married and have children'
have children/raise a family

If you insist ion going abroad now, you will be too old to complete thesmight miss out on starting a family later in life."

I wrote it as I'd say it!

I said nothing, because I knew that itdoing so would be just be a waste of time.

'I remained silent, because I knew that I wouldn't be able to change her mind.'

She thinks I athat I'm stubborn, so do Iand to be honest I agree with her.