Jalebi. I feel inadequate to the task and hope RV or PSquared jump in soon because they are the advisers supreme on this board.
I'll do my best, however.
I agree that the way your mother handled it - not telling your father the time, deliberately going out when she knew what time they would call - was not a good method.
You have spoken to your parents. Calmly and non-calmly, as you've said. And as you've admitted, your mother has some true concerns. Now you need to sit down and think about this some more in light of your mother's VALID objections...emphasis on valid as some of her other objections (height, distance, ethnicity) aren't so valid. If after thinking about it and doing Istikhara, you still have your heart set on Q, then I think it's time to involve another trusted elder. Speak to an aunt, a nani, an Imam. Explain what has happened calmly and in an organized fashion. And ask for that person's help in talking to your parents (and possibly to Q and his family).
If Q is the one you want as your spouse, then it looks like you're in for a fight. But don't give up. Right now, I think you want to, but trust me...you'll hate yourself later if you do. Nothing that's worth having is easy. There are always obstacles blocking you from the thing you want most. Unfortunately, it looks like it may be your parents in your case. So remember that they love you, be respectful...but keep fighting. If it doesn't work out, it doesn't work out. Inshallah, that too shall pass. But at least when you look back on it, you can at least reassure yourself that you did your best.