The accepted answer gives the correct number, but is incomplete. It's true that Black can always force either 3. ... Q(x)d2+ or 3. ... Q(x)f2+. However, what if White could choose a fourth move that is valid even if such a check is made? The only possible options are 4. Kd1, 4. Kf1, 4. Kd2 or 4. Kf2. It turns out that Black can still prevent such sequences, but for various different reasons that don't necessarily involve one of these checks.
First, White needs three moves to prevent all the routes to Qf2+, and none of these can be bishop moves so 4. Kf1 is out. If White moves the f pawn and makes two other moves to block Qf2+ Black can always play Qxf3 or Qxf4, so 4. Kf2 is also out (White doesn't have time to play Nf3 here without allowing Qf2+). Note that, even if you consider 4. Kxf2 and 4. Kf2 to be the same move, White must block Qf2+ here because otherwise Black can just leave the queen attacking f2 to prevent Kf2.
So does White have a sequence ending 4. Kd1 or 4. Kd2? To do this, White must use two moves to prevent Qd2+. This means there is no time to get anything on c5, and so White can't avoid 1. ... e3, 2. ... Bb4, 3. ... Bc3, (where the bishop moves might be captures) which prevents 4. Kd2.
The only option remaining is 4. Kd1 and that requires two moves to prevent both routes to Qd2+, and one queen move to vacate d1. The non-queen moves need to be (b4 or c3 or Nc3) and (e3 or f4). However, f4 is not possible: Black can play 2. Qh4, which is either a check preventing White's intended third move or prevents 3. f4.
If White plays e3, Black's plan is to avoid 4. Kd1 by getting a bishop to the d1-h5 diagonal. If the White queen is on this diagonal after White's third move, Black can always capture with the bishop on move 3. If not, Bg4 is sufficient to prevent 4. Kd1, and White can't stop this while also stopping Qd2+ and moving the queen.