Why does Azo coupling of β-naphthol takes place at alpha position and not at gamma position
Coupling reaction of β-naphthol with benzene diazonium is an example of electrophilic aromatic substitution.

If the electrophile attacks at alpha position ,then two resonance structures 1 and 2 , with aromatic rings are possible.

 
If the electrophile attacks at gamma position ,only one resonance structures 3 , with aromatic ring is possible and 4 is not aromatic.



Therefore attack at alpha position is the major product.

References:

https://pubs.rsc.org/en/content/articlehtml/2015/gc/c4gc02381a

https://en.wikipedia.org/wiki/Azo_coupling

Mechanism of formation of 2-naphthol red dye (aka Sudan 1)