The given problem 4yk(k+1)+1=a^2 leads to (a^2-1)=y(b^2-1), which has infinitely many solution, but to give all of them is difficult. One can specify classes providing infinitely many of them, for example

1. a=b^(2u), or

2. when y is not a square, then a^2-yb^2=1-y is a generalized Pell equation possessing again infinitely many sols, since it is soluble: a=1=b.

Yes, I know. It is difficult to explore all solutions.