Hi, these sentences have different meanings. I have rephrased them below :
The 1st sentence, "I didn't ask her for more details, I believed what she said was true without looking into it."
The 2nd sentence, "I didn't ask for more details, I didn't care enough to learn more."
The 1st sentence expanded could be rephrased to be, "She is a doctor so I didn't ask her for more details about her diagnoses, I just took it at face value (Took her for her word / professional opinion)."
The 2nd sentence expanded could be rephrased as, "My coworker told me during the meeting she was looking into buying a Toyota. Our manager walked in right at that moment so I didn't ask for more details, I just let the topic drop."
I hope this helps!