A CS professor yesterday asked me this query. I think there is no direct relation if any.
The frobenius number is the largest number that cannot be represented by $au+bv$ where $gcd(a,b)=1$ holds and $a,b,u,v\in\Bbb N$ holds.
Elliptic curves are isomorphic over $\Bbb C$ to $2$ dimensional lattices which at least superficially looks like $au+bv$ forms.
So is there any reason to expect no connection between elliptic functions and linear forms in $2$ variables?
Yes, there is a connection between elliptic curves over finite fields and the Frobenius Map, which is the function $$ \tau_p \colon E(\overline{\mathbb{F}}_p)\rightarrow E(\overline{\mathbb{F}}_p),\quad \tau_p(x, y) = (x^p, y^p). $$ One can check that $\tau_p$ is a group homomorphism. It gives the quantity $$ a_p=p+1-|E(\overline{\mathbb{F}}_p)|, $$ called the trace of the Frobenius, which satisfies $|a_p|\le 2\sqrt{p}$ by Hasse.
I think that "Frobenius number" is not directly connected to elliptic curves, so you are right.