The question is not strictly well-defined. But it goes like this:

Could you find an infinite family of graphs $G_i$, that are all $\epsilon$-expanders, but have many 4-cycles?

$\epsilon$ should be as large as possible as well as the number of 4-cycles. The family of graphs should be as wide as possible.

This is the basic question.

A trivial good example would be a cartesian graph product $G \times H$. The expansion of the product is the maximal $\epsilon$ of the involved graphs. If we take two $d$-regular graphs and we look at 2 paths of the product, then about $1/3$ of the paths are in $4$-cycle.

But could it be improved? Could the relation be bounded?

If we think in terms of Cayley graphs:

Say our graph is $\mathrm{Cay}(G,S)$ and $S$ is symmetric. A $4$-cycle arises from $4$ generators $a,b,c,d$ s.t. $ab=cd$. It is trivial if either ($c=b$ and $d=a$) or ($c=a$ and $d=b$).

I think many trivial cycles would harm expansion since, by Alon-Roichman theorem, abelian groups are bad expanders.

So I think the goal should be to find groups and generator sets that have many non-trivial 4-cycles.