MPs voted by 432 votes to 202 to reject the deal, which set out the terms of Britain's exit from the EU on March 29.
Mrs Trevelyan, a committed Brexiteer, joined those voting against Theresa May's Government.
Outlining her reasons, she said: "I have received hundreds of emails from constituents on this issue, with views split approximately 50-50 on whether I should back the deal or not.
"I have been clear throughout the negotiation process that I was willing to be pragmatic, and although there were many elements of the proposed deal I found unpalatable, I would have been prepared to agree to them in order to achieve a smooth exit from the EU.
Police seize large drugs haul in north Northumberland
Wildlife sightings cause excitement on Northumberland Wildlife Trust's East Chevington nature reserve
Communities come together to remember Battle of Flodden
Liz Truss and Rishi Sunak make A1 dualling pledge if made Prime Minister
New caravan site gets the green light on the Northumberland coast
"However, along with many colleagues from all sides, the Northern Ireland backstop caused me great concern, because if triggered it would mean remaining within the EU customs union as rule-takers, with no exit mechanism for an indefinite period of time.
"I have spent the past few weeks meeting with the Prime Minister and others and she understands why that element causes problems, and pledged to go away to find a legally-binding solution.
"No such solution has yet been proposed, as such I could not support the deal in the vote this week. I continue to remain hopeful the Prime Minister will be able to keep the promises she has made on that issue."In the meantime I am working with various Government departments who are ensuring there would be as smooth a transition as possible in the event of no deal having been agreed when we exit on 29th March.
"For example, the Health Secretary is very confident that all medical supply concerns have been addressed, cross-border arrangements for SMEs are being addressed, transport issues are being finalised and I have also received firm assurances from the Environment Secretary that his Department has contingency plans for lamb tariffs which are essential for our Northumbrian farmers.
"If constituents have specific concerns about things which they believe could affect them in such a no deal scenario, they should raise them with me as soon as possible."