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.
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.