Should the Quality Department originate a Request for Deviation to the Customer?

Should the Quality Department originate a request for deviation to the customer or would it be considered a "conflict of interest"?

It depends! If not specified in your internal SOP's the consideration of which functional group will "originate" should:
- meet the requirements of the contract with the customer (this is a "must")
- "originate" might actually be preparation and then submission - two sub-processes requiring different collaboration from functional groups to result in sending to your customer
- provide convenient, efficient, effective single point of contact for your customer to respond back or engage in discussion

Quality Dept should be on distribution for sure.

Requesting a deviation from the customer {I assume} means that you aren't going to follow a mutually agreed requirement.

Customer relationship manager (sales, engineering, other} would be my first pick...together with the person who can best make the case for what, why, how long and potential impacts.

If you aren't going against mutually agreed items, why involve the customer at all?

At the end of the day, it should be done by the person who can succeed in getting approval without irking the customer...regardless of position or title.

If the organization engages Quality in the contract review process, and they discover a need for the deviation request, perhaps they are the best to communicate it.

I would gage what would be the most effective method.