(Not an answer; just a comment.)

It is a somewhat tangled graph. Here is a representation of $G(100)$:
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![SqGraph100][1]
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One can see $50+50=100=10^2$ at the lower-right corner,
$69+52=111=11^2$ near the upper-right corner,
$70+51=121=11^2$, left midlevel, etc.

This $G(100)$ graph has diameter $5$. But $G(1000)$ has diameter $4$.


  [1]: https://i.sstatic.net/MA3hx.jpg