Various variants of the matrix $$A_{J,K} = |J\cap K|$$ were studied, and the spectrum was computed. A one-parameter variant is given by $$A^{(i)}_{J,K} = \binom{|J\cap K|}{i}$$ for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in a this friendly paper by Ghareghani, Ghorbani and Mohammad‐Noori. In particular, the spectrum of $A^{(i)}$ is described in Lemma 9. The lemma is attributed to R. M. Wilson (1982), who uses somewhat different notation and terminology.
Other useful references, including to additional works of R. M. Wilson, appear in the bibliography of the G-G-MN paper.