$\newcommand\al\alpha$If the $(1-\al)$-quantiles of $F_{n,m}$ were decreasing monotonically in $m$ for each $\al\in(0,1)$, the the corresponding cdf's would be increasing pointwise on $(0,\infty)$ monotonically in $m$. However, this is not so, as seen from the graphs below of $G_{3,1}-G_{3,1}$ (red), $G_{3,2}-G_{3,1}$ (orange), $G_{3,3}-G_{3,1}$ (green), $G_{3,4}-G_{3,1}$ (blue), $G_{3,5}-G_{3,1}$ (magenta):

However, recall that $F_{n,m}$ is the distribution of $$R_{n,m}:=\frac{X_n/n}{Y_m/m},$$ where $X_n$ and $Y_m$ are independent chi-squared random variables with $n$ and $m$ degrees of freedom, respectively.

By the law of large numbers, $Y_m/m\to1$ in probability as $m\to\infty$. So, as $m\to\infty$, $F_{n,m}$ converges weakly to the distribution of $X_n/n$, which is the gamma distribution with parameters $n/2,2/n$.

$\newcommand\al\alpha$If the $(1-\al)$-quantiles of $F_{n,m}$ were decreasing monotonically in $m$ for each $\al\in(0,1)$, the the corresponding cdf's -- say $G_{m,n}$ -- would be increasing pointwise on $(0,\infty)$ monotonically in $m$. However, this is not so, as seen from the graphs below of $G_{3,1}-G_{3,1}$ (red), $G_{3,2}-G_{3,1}$ (orange), $G_{3,3}-G_{3,1}$ (green), $G_{3,4}-G_{3,1}$ (blue), $G_{3,5}-G_{3,1}$ (magenta):

However, recall that $F_{n,m}$ is the distribution of $$R_{n,m}:=\frac{X_n/n}{Y_m/m},$$ where $X_n$ and $Y_m$ are independent chi-squared random variables with $n$ and $m$ degrees of freedom, respectively.

By the law of large numbers, $Y_m/m\to1$ in probability as $m\to\infty$. So, as $m\to\infty$, $F_{n,m}$ converges weakly to the distribution of $X_n/n$, which is the gamma distribution with parameters $n/2,2/n$.