we have (p^2-q^2)^2=(p-q)^2(p+q)^2

so we can immediately get some generalized by haveing

p-q=16q+1 -> p=17q+1

p-q=-16q-1 -> p=-15q-1

p+q=16q+1 -> p=15q+1

p+q=-16q-1 -> p=-17q-1

thus we have that

q>=0 p=+-(17q+1)  p=+-(15q+1) generates solutions although not neccesarily all solutions.