Consider the problem of a classical pendulum whose state can be described by a function $\theta(t)$ where $\theta$ is measured from the line directly below. We then have that our pendulum's $\theta$ obeys the following differential equation
$$ \frac{d^2 \theta}{dt^2 } + \frac{g}{l}\sin \theta = 0 $$
Through the trick $ K = \frac{d\theta}{dt}, K \frac{dK}{d\theta} = \frac{d^2 \theta}{dt^2}$ we can re-write the above differential equation as a different one and then integrate it to find that there is a constant $Q_0$ such that
$$ \frac{1}{2} \left( \frac{d \theta}{d t} \right)^2 – \frac{g}{l} \cos(\theta) = Q_0 $$
It's fruitful to ask "what does this really mean?", what is that $Q_0$ actually supposed to be? and by multiplying both sides by $ ml^2 $ we find rather enlightening that we have the following:
$$ \underbrace{\frac{1}{2} ml^2 \left( \frac{d \theta}{d t} \right)^2}_{\text{Kinetic Energy}} + \underbrace{-mgl \cos(\theta)}_{\text{Gravitational Potential Energy}} = ml^2 Q_0 = E_0 $$
And now this is much less mysterious, it is clear this $Q_0$ is just a scaled version of $E_0$ the total energy of our system, which is constant as we should expect. Of course we can continue going forward here… Before we added the extra mass-length information the differential equation could have been re-written as:
$$ \frac{1}{\sqrt{2Q_0 + \frac{g}{l} 2\cos(\theta)}} \frac{d \theta}{d t} = 1$$
Again this can be integrated to yield another quantity…
$$ \sqrt{\frac{2}{Q_0 + \frac{g}{l}}} F \left[ \frac{\theta}{2} , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right] = t + Q_1 $$
This suggests then that the following is true…
$$ \sqrt{\frac{2}{Q_0 + \frac{g}{l}}} F \left[ \frac{\theta}{2} , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right] -t = Q_1 $$
I.E. there is some quantity $Q_1$ which does NOT vary with time, and can be found through that horrendous looking left hand side. What conserved Quantity is this $Q_1$ supposed to represent kinematically? It should be something akin to a "Second Energy" or "Momentum" of our pendulum but I can't figure what this thing is supposed to be and there doesn't seem to be any descriptions of it online. It does appear to be intimately related to the period. One could also theoretically verify it is conserved by measuring that LHS in an experiment and confirming it doesn't vary with time.
Some realization:
If you declare the state of your system to be $S$ at time $t=0$ then at any time thereafter you would also declare that "back in $t=0$ the state was $S$". The 'conservation' of $Q_1$ appears to be a restatement of just that.
Best Answer
Summary
$Q_1$ is not a conserved quantity at all. It is just a parameter which depends on the initial conditions.
Error
First of all, there's an error when you derived
$$\frac{1}{\sqrt{2Q_0 + \frac{g}{l} 2\cos(\theta)}} \frac{d \theta}{d t} = 1\tag{1}$$
You only took the positive square root, whereas you should have take both the possibilities of the RHS being $+1$ and $-1$. You can easily see that the equation $(1)$ never holds whenever $\theta$ is decreasing i.e. $\mathrm d \theta /\mathrm d t<0$. To correct this, we need to add a modulus around the $\mathrm d \theta/\mathrm dt$ term. Thus the corrected equation would be
$$\frac{1}{\sqrt{2Q_0 + \frac{g}{l} 2\cos(\theta)}} \left|\frac{d \theta}{d t}\right| = 1\tag{2}$$
I would advise you to re-integrate equation $(2)$ to find the correct solution which holds over the complete range of motion.
What about the other equation
The final equation which you obtained
$$\sqrt{\frac{2}{Q_0 + \frac{g}{l}}} \left(F \left[ \frac{\theta}{2} , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right]\right) -t = Q_1\tag{3}$$
only holds true for the cases where $\mathrm d \theta/\mathrm dt>0$, so for now, we'll only consider cases where the pendulum is going from left to right, but the insight provided below will also help you determine the physical meaning of the new constant that you would obtaing after integrating equation $(2)$. Also, the equation $(3)$ contains an incomplete elliptic integral of the first kind. One of the important properties of this function is that
$$F[0,k]=0$$
where $k$ is any real number. Thus, substituting $\theta=0$ in equation $(1)$, we get
\begin{align} \sqrt{\frac{2}{Q_0 + \frac{g}{l}}} \left(F \left[ 0 , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right]\right) -t_0 &= Q_1\\ 0-t_0&=Q_1\\ Q_1+t_0&=0\tag{4} \end{align}
where $t_0$ is the time when the pendulum passes throught its equilibrium position for the first time. And since we are only considering the case where $\mathrm d\theta /\mathrm dt>0$, thus the above equation is valid only for the cases where the pendulus comes from the left and goes to the right while passing through the equilibrium position.
Physical Significance
The physical significance of the constant $Q_1$ isn't as deep and profound as you expected. $Q_1$ is just a shifting constant applied to the time. This constant will change upon changing your definition of $t=0$. Thus, it's just a parameter which adjusts/shifts the time scale of the oscillation. It adjusts according to the initial conditions and doesn't give you any more information about the dynamical parameters of the system.