Well, it is a cheap threat for a beginner and I wouldn't have a problem with it on its own merit. However, it seems that the queen is misplaced if White is playing for d2-d4 as it doesn't defend that square anymore. It blocks the f pawn which is White's plan to advance for activity. It's not THAT bad, certainly playable, but certainly dubious compared to other openings. I remember when I first started I played the French without d5 in order to avoid the four move checkmate.
Napoleon's Opening

I'm a bit late to the party but I've experimented with 1. e4 e5 2. Qf3 in blitz and tournament play, but with a different plan. I don't think plans involving Bc4 work well, the threats are too obvious. There's a story behind how I started (occasionally) playing this move.
When I was in high school, a certain eccentric Ohio master was playing this line - even against the state champ - on occasion. I asked him about it and he explained that it was being analyzed by members of the Cincinnati or Dayton chess clubs (can't recall which, this was like close to 25 years ago!).
The idea was to play a kind of light-squared Stonewall attack with pawns on c3,d3, e4, g4, and h3 against black's classical center, with knights on d7 and e7, with the pawns preventing annoying attacks on the Queen, and with the Queen herself helping to enhance White's light-square control. F5 in particular becomes an important square, with a white Knight frequently occupying it and forcing an exchange, thereby opening up the g-file for White's R. The King often remains in the center, but it is generally safe there if you are careful to cover the critical squares d3, d2, and d1 well.
The setup is very solid and leads to interesting play. The provenance of the line is rather mysterious. My Ohio friend claimed it was an old Tartakower idea, but I never found anywhere where Tarta mentioned it, and no games where he played it. However, William Lombardy and Julian Hodgson experimented (as Black) with a Philidor line 1. e4 e5 2. Nf3 d6 3. Bc4 Qf6!? and played essentially the same idea. After a lot of success in blitz, 3. ...Qf6 became my standard tournament response to 3. Bc4.
( I present this not as a great game, but just to illustrate White's setup).
Here, too, is a more competent player's win (this one playing the setup as Black!)
I began to take a "in-head" visual of Napoleon's Opening. I told my
wife that I could teach her a 4-move checkmate – most people who have
studied chess are aware of this opening. It normally involves 1. e4
e5 2. Qf3, also known as Napoleon’s Opening, but you could also play
1. e4 e5 2. Bc4, the Bishop’s Opening.
Of course, the basic idea involves e4, Bc4, Qf3 (or Qh5, but then the
queen is in a different area of the board and the opening dynamics are
changed) and aiming for Qxf7#. Sure, I know – these are simple
schemes. But I debated to myself – if I were to teach my wife this
opening, and they defend against it correctly, she ought to know a
simple way to develop.
Of course there will be scoffers out there – “Don’t teach her that,
it’s incorrect!” These will be the same people who maybe never had
the joy of playing the four-move checkmate as a beginner – they were
probably the ones who got burned by it when they first begun – that
was a joke by the way.
My point behind chess is that it should be fun, and I know it is fun
to be a beginner and to have pulled off a four-move checkmate
successfully.
I know with my experience in different openings, I have not really
looked into the opening dynamics of Napoleon’s Opening until this
morning, and just a bit – so I want to share my tiny bit of epiphany
with you and see what kind of feedback we can get here. I am already
prepared for the barrage of “this is dumb” ideas… so if that is the
best you have, please don’t waste anyone’s time. This is for someone
who may have a fun time trying an offbeat opening.
First, before I get straight into the Napoleon, I want to share a bit
of a Danish Gambit I played with a quick Qf3. 1. e4 e5 2. d4 exd 3.
c3 dxc 4. Bc4 cxb 5. Bxb2. I don’t remember exactly what happened…you
can research my games if you want, but I did play an early Qf3, my
opponent missed the threat, and I played Qxf7#. Satisfying? Maybe
for a second. It was an elementary threat, and the satisfaction
didn’t come from gaining a cheap win – it came from my experiment in
the Danish Gambit.
But I still remember that elementary win – the game is burned into my brain.
I remember being beat many times when I was first faced by this opening from the black side. I learned chess originally as a e4 and ...e5 player, and so he would play Qf3 and Bc4 and Qxf7# time and time again...I didn't understand Nf6, I guess I was afraid he would take my knight with the queen! lol Funny things you do when you are learning.
So anyway, here we are with the moves 1. e4 e5 2. Qf3 and let's say that he plays 2...Nf6. The threat of Qf7 is gone, but can White continue with his ideas? If 3. Bc4 and then ...Nc6, then c3 stops threats of Nd4 and Nb4, prepares to push the pawn to d4, and the bishop has a comfortable cubby to retreat to.
Note - I am not saying that you should try this against a GM - I am simply talking about a beginner exploring this opening. Who knows, I may try it myself sometime just for fun.
Now, the king knight can proceed with Nge2, and White is ready to castle. If the queen needs to move, she can go to g3 and prepare either f3 or f4, and White's queen knight can either come out via a4 and Na3, or Nbd2. White can wait to play his queen bishop until the right moment...so b2 is covered. Also, with the queen on f3, White can look into ideas involving h3 and g4, or even a direct g4 attack on Black.
You may consider trying this in blitz for fun someday - I don't know. My idea that I spotted this morning that I am going to look into is 1. e4 e5 2. Qf3 Nf6 3. Bc4 Nc6 4. c3, in an Italian Game fashion, but with the queen on f3. Now I am going to go to Game Explorer and see results. Notice - I never like to look at results before I do this, because then original thought is lost.
Now, there is something for you to chew on!