Can anyone tell me if the part which is on quotes is correct or if I can use another words to express the same.
She left "without heeding my words", she just ignored my warning.
Yes, it is correct. However, what you have written looks like it should be two separate sentences:
"She left without heeding my words. She just ignored my warning."
I cannot think of a substitute for what you have marked in quotations, but I can substitute the whole sentence:
"I knew she wouldn't do as I said."
"I knew she wouldn't listen to me."
"She refused to listen to what I was saying, and left."
Or maybe it's enough to say "She just ignored my warning", as it means very much the same thing as the first sentence.
She left without paying attention,she just ignored my warning.