Can we establish a relation between the distribution functions of $F_{m,n}$ distribution and Binomial Distribution?
I have a vague idea that if $X \sim \text{Bin(n,p)}$ we can write $P[X \le k]=\sum_{x=0}^{k} {n \choose x} p^x (1-p)^{n-x}=I_{1-p}(n-k,k+1)$ where $I_x$ is the incomplete Beta function. Now we also know that if $\frac{X}{Y} \sim \text{Beta}_2(\frac{m}{2},\frac{n}{2})$,then $\frac{nX}{mY} \sim F_{m,n}$. Where, $\text{Beta}_2$ is the beta 2nd kind distribution. How will I relate the two? Help needed!
Yes, there is a known relationship between the Binomial and F distributions.
This is documented in this paper by G.H. Jowett.
But we can derive the relationship using elementary properties of known distributions.
If $X$ is a binomial variable with parameter $(n,p)$ and $F_{m,n}$ is an $F$ statistic with $(m,n)$ degrees of freedom, then one can establish the following identity involving their distribution functions:
$$P(X\leqslant k-1)=P\left(F_{2k,2(n-k+1)}>\frac{n-k+1}{k}\cdot\frac{p}{1-p}\right)\quad,k\in\mathbb N\tag{*}$$
You correctly noted that $F_{m,n}$ can be expressed as $F_{m,n}=\frac{n}{m}\cdot\frac{U}{V}$ where $\frac{U}{V}\sim\beta_{2}\left(\frac{m}{2},\frac{n}{2}\right)$, the beta distribution of the second kind (or the beta prime distribution).
Denoting the beta distribution of the first kind by $\beta_1$, this means we have \begin{align}&\frac{V}{U}\sim\beta_{2}\left(\frac{n}{2},\frac{m}{2}\right)\\&\implies \frac{\frac{V}{U}}{1+\frac{V}{U}}\sim\beta_1\left(\frac{n}{2},\frac{m}{2}\right)\\&\implies \frac{\frac{n}{mF_{m,n}}}{1+\frac{n}{mF_{m,n}}}\sim\beta_1\left(\frac{n}{2},\frac{m}{2}\right)\\&\implies\left(1+\frac{mF_{m,n}}{n}\right)^{-1}\sim\beta_1\left(\frac{n}{2},\frac{m}{2}\right)\end{align}
It immediately follows that the CDF of $F_{m,n}$ is given by $ P(F_{m,n}\leqslant x)=I_{mx/(mx+n)}(\frac{m}{2},\frac{n}{2})$
As you noted, $P(X\leqslant k-1)=I_{1-p}(n-k+1,k)$ where $I_x$ is the regularized incomplete beta function. So you have the l.h.s of $(*)$. Using the expression for the CDF of $F_{m,n}$, after a little simplification, it follows that the r.h.s is also $I_{1-p}(n-k+1,k)$.
A property of the incomplete beta function that will be required is that $I_x(a,b)=I_{1-x}(b,a)$.