1. I inquired her a question.
2. I inquired a question of her.
Can inquire take two objects like sentence 1?
Are the both grammatical?
Neither sentence is correct. We don't use "a question" with "inquire/enquire".
I asked her what time the last train leaves.
I enquired of her what time the last train leaves.
"What time does the last train leave?",I asked her.
"What time does the last train leave?", I enquired of her.
Effectively, "to enquire/inquire" means "to ask a question" so it would tautology to repeat "a question".