A: Why did you call me last night?
B: I just wanted to know if I could use your car when/whenever my girlfriend asks me to pick her up at her home.
I would like to know if the verbs in bold are in the correct tense.
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In general you shouldn't switch from one tense to another unless the timing of an action demands that you do.
I just wanted to know if I could use your car when/whenever my girlfriend asks me to pick her up at her home.
In your dialogue,
Person B establishes the past-tense. The verbs
Person A uses: wanted
and could are both past-tense (so correct). The verb asks is not, but this is still correct because it is preceded by when/whenever.
This means that
Person A's girlfriend is subject to ask(s) at any time (including the past, present, and future); thus, it is correct. If you were to use the past tense (asked), that would be incorrect because it doesn't cover the present and future.