According to here, the harmonic conjugate of a harmonic function $u$ is given by
$$v(z)=\int_{z_0}^z u_xdy-u_ydx+C$$ where $C$ is a constant, while in here, the harmonic conjugate is given by
$$v(z)=\int u_xdy-\int u_ydx-\iint u_{xx}dxdy$$ where the integral is an indefinite integral. The last term above is no way a constant (since it is an indefinite integral). I am wondering whether there is any relation between the line integral and the indefinite integral.
In conclusion, my question is: why are the above two formulas equivalent?
It is not clear to me what is actually meant by the second formula. More precisely, what is $$\phi(x,y) =\int \int f(x,y) d\,x d\,y $$ supposed to mean? Note that on both sides $x$ and $y$ are used for the variables, but in a totally different context. Once you fix the domain on the rhs, it does not even depend on $x$ or $y$.
The author of the text your second link is pointing to is rather sloppy with regard to dependend and independent variables.
I'm also under the impression that there is a mistake in his reaoning. In equation 3-27 he claims that $$v(x,y) = \int v_x(x,y) d\, x + C(x)= \int v_y(x,y) d\, y + C(x) $$ with a function $C$ (a single function!) depending on $x$ alone.
Ignoring for a moment what these integrals might mean let's look a bit closer at what is actually true. Note that $$v(s,t) =\int_{s_0}^s v_y(x,t) d\, y + v(s_0,t) = \int_{t_0}^t v_x(s,y) d\, x + v(s,t_0)$$
(assuming the domain on whic $v$ is defined allows to write down these integrals).
This shows that, when you integrate wrt $x$ the assumed function $C$ depends on the second free variable while, when you integrate wrt $y$, that function will depend on the first variable, and the two functions will by no means be equal to each other.
The claim in 3-27, therefore, is not correct in the way it is stated.
I won't answer the question about the formula you are after, but I would not know what the second formula actually means and there is at least one error in the attempt to derive it, so I would not trust it.
Rather, I suggeest to try to figure the formula out yourself, with some more care paid to the difference between dependent and independet variable.