My personal opinion is no. Why ?
First a definition of prorogation (from the wikipedia entry on Parliamentary Session(My emphaisi in bold text)):
"A prorogation is the period between two sessions of a legislative body. When a legislature or parliament is prorogued, it is still constituted (that is, all members remain as members and a general election is not necessary), but all orders of the body (bills, motions, etc.) are expunged....
In the ... Canadian parliamentary systems, this is usually due to the completion of the agenda set forth in the Speech from the Throne ..."The F.U. is the very first business put before the House of Commons. They have obviously lost the confidence in the House.
"If a Cabinet is defeated in the House of Commons on a motion of censure or want of confidence, the Cabinet must either resign (the Governor General will then ask the Leader of the Opposition to form a new Cabinet) or ask for a dissolution of Parliament and a fresh election." (Notice which option is listed first - C.V.)
"In very exceptional circumstances, the Governor General could refuse a request for a fresh election. For instance, if an election gave no party a clear majority and the Prime Minister asked for a fresh election without even allowing the new Parliament to meet, the Governor General would have to say no. This is because, if “parliamentary government” is to mean anything, a newly elected House of Commons must at least be allowed to meet and see whether it can transact public business. Also, if a minority government is defeated on a motion of want of confidence very early in the first session of a new Parliament, and there is a reasonable possibility that a government of another party can be formed and get the support of the House of Commons, then the Governor General could refuse the request for a fresh election. The same is true for the Lieutenant-Governors of the provinces."Let us read one part again: if “parliamentary government” is to mean anything, a newly elected House of Commons must at least be allowed to meet and see whether it can transact public business. It must be allowed to meet and see whether it can transact public business.
From the post itself:
“It does not mean that the Crown should act arbitrarily and without the advice of responsible ministers, but it does mean that the Crown is not bound to take the advice of a particular minister to put its subjects to the tumult and turmoil of a series of general elections so long as it can find other ministers who are prepared to give it a trial. The notion that a Minister – a Minister who cannot command a majority in the House of Commons – is invested with the right to demand a dissolution is as subversive of constitutional usage as it would, in my opinion, be pernicious to the general and paramount interests of the nation at large.”
Marshall, Geoffrey Constitutional Conventions: The Rules and Forms of Political Accountability. Oxford, 1984, p. 38.
So he doesn't have the right to ask for dissolution. He almost certainly does not have the confidence of the House of Commons. The only purpose in granting a prorogation would be to enable a coward in an attempt to avoid his fate during a crisis.
She should refuse prorogation and request that the House meet ASAP to vote on the non-confidence motion.
Update: And another thing. He has been daring the Opposition to take him down for three years now. The result has been policies a strong majority disagreed with being forced on us due to the Opposition respecting the electorate's expectation that a minority Parliament will work together. Well, they finally called his bluff.