When we talked about my going abroad, she just said 'Do you think you can go out?'.
I'm not sure I understand this. Was she asking if you will be able to get the necessary permission to leave the country?
Maybe as a good friend, I must consider these as a kind of...greeting, or concerning? But there is a meaning of despising, or worse, being jealous, isn't it?
I can't imagine why, from what you've written here, you would suspect jealousy. Is there some evidence of that feeling that you've not written here? What you've reported so far is a normal conversation between friends:
"I hope to go abroad."
"Do you think you'll be able?"
"I have to take a hard test."
"Do you think you will pass?"
And when she asked about my private things, such as grades, as I'm modest, I didn't tell her the truth to avoid being regarded as showing off.
If you've not seen each other for some years, she probably asked about your grades either as a friendly general inquiry or as part of the conversation above:
"I have to take a hard test."
"Do you think you will pass? What are your grades like now?"
It sounds like you know there is some problem in how you interpreted her questions and that is what troubled you enough to inspire you to write this essay. For the writing, and perhaps for your friendship too, the thing to do is go deeper, to write in more detail about the interaction and what preconceptions
you brought to it. Do you tend to feel that people are jealous of you? Has that interfered with friendships before? What changes might you make to avoid that happening again? In conversations, would it be possible for you to simply ask someone what they mean, rather than assuming something or wondering about it afterwards?