Here I've asked what is a quotient of an object and the answer was that it is an equivalence class of epis.
But here on the first page they claim that regular quotient is an coequalizer of 2 morphisms. I do not follow why this coequalizer should be epi.
It seems that your question is: If $c\colon B\to C$ is the coequalizer of the pair $f,g\colon A\to B$, why is $c$ an epimorphism?
To show that $c$ is an epimorphism, suppose $p,q\colon C\to D$ are two arrows such that $p\circ c = q\circ c$. We need to show that $p = q$.
Let $h = p\circ c = q\circ c\colon B\to D$. Since $c$ coequalizes $f$ and $g$, we have $c\circ f = c\circ g$, so $h\circ f = p\circ c \circ f = p\circ c \circ g = h\circ g$. Thus $h$ coequalizes $f$ and $g$. By the universal property of the coequalizer, there is a unique arrow $h'\colon C\to D$ such that $h'\circ c = h$. By uniqueness, $h' = p = q$.
An arrow is called a regular epimorphism if it is a coequalizer for some parallel pair of arrows. Dually, every equalizer is a monomorphism, and an arrow is called a regular monomorphism if it is an equalizer for some parallel pair of arrows.