How to extend background fill color scope for a tree?

You could just extend the fit by the shadow xshift and shadow yshift. This is what is done with the shifts in

(current bounding box.north west) rectangle 
([yshift=-0.5ex,xshift=0.5ex]current bounding box.south east);

Then, your tree contains some shifts of the nodes, which where probably adequate for an earlier version. In order to find out which shift you need, remove the

before drawing tree={x+=36.06616pt}

key from the root node and uncomment

\path let \p1=($(dep)-(pres)$) in \pgfextra{\typeout{\x1}};

This will tell you the precise dimension by which you need to shift.

\documentclass{article}
\usepackage[margin=1in]{geometry}%<- to make the page fit the tree
\usepackage{tikz}
\usepackage{tikzpagenodes}
\usetikzlibrary{fit,shadows,fadings,backgrounds}
\tikzfading[name=fade right,
left color=transparent!0,
right color=transparent!100]
\usepackage{fontawesome5}
\usepackage{xcolor-material}
\usepackage[sfdefault]{FiraSans}


\colorlet{cinza}{MaterialGrey600}
\colorlet{vermelho}{MaterialRed400}
\usepackage[edges]{forest}

\begin{document}

        \begin{forest}
        for tree={anchor=center,%<-added
            fill=cinza,rounded corners,align=center,rectangle,
        drop shadow,text=white,inner sep=6pt,l sep=.6cm,
        fork sep=.3cm,
     %line width=1pt,
    %draw=MaterialYellow800,
     edge={thick, draw=cinza, rounded corners=1pt},
},
for 1={fork sep = .6cm},
for 3={fork sep = .6cm},
 forked edges,
 [Presidente, l sep+=2em,before drawing tree={x+=36.06616pt},fill=MaterialOrange600,alias=pres
        [Departamento\\Comercial
            [Ger. Marketing]
            [Ger. Vendas]
            ]
 [Departamento\\de Produc\~ao,alias=dep
 [Ger. Produc\~ao]
 [Ger. Compras]
 ]
[Departamento\\Financeiro,before drawing tree={x+=2.5cm}]
]
% restore this if you need to readjust the position of the root node
%\path let \p1=($(dep)-(pres)$) in \pgfextra{\typeout{\x1}};
\begin{scope}[on background layer]
    \fill[top color=MaterialGreen400,bottom color=MaterialGrey200,inner ysep=1em, inner xsep=1em, outer sep=1pt,%path fading=fade right
    rounded corners
    ] 
    (current bounding box.north west) rectangle 
    ([yshift=-0.5ex,xshift=0.5ex]current bounding box.south east);
\end{scope}
\end{forest}

\end{document}

If you make the bounding box more generous, e.g. by using

\fill[top color=MaterialGreen400,bottom color=MaterialGrey200,inner ysep=1em, inner xsep=1em, outer sep=1pt,%path fading=fade right
rounded corners
] 
([xshift=-1em,yshift=1em]current bounding box.north west) rectangle 
([yshift=-0.5ex-1em,xshift=0.5ex+1em]current bounding box.south east);

you'll get

This is a hand-made shadowed node by plain TikZ.

\documentclass[tikz,border=0mm]{standalone}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}[n/.style={rounded corners,align=center,minimum height=1.2cm,minimum width=3.5cm,font=\sffamily,text=white}]
\def\shadowednode#1#2#3#4{
\path #1 node[n,fill=gray!50,shift={(.2,-.15)}]{#2}
node[n,fill=#4] (#3) {#2};
}
\shadowednode{(0,0)}{University of \\Natural Sciences}{NS}{orange}
\shadowednode{(0,-2.5)}{Department of\\ Mathematics}{M}{brown}
\shadowednode{(5,-2.5)}{Department of\\ Physics}{P}{brown}
\shadowednode{(-5,-2.5)}{Department of\\ Biology}{B}{brown}

\draw 
(NS.south)--(M.north) 
(NS.south)--++(-90:.5)-|(B.north)
(NS.south)--++(-90:.5)-|(P.north);
\begin{scope}[on background layer]
\fill[top color=green!50,bottom color=teal]
(current bounding box.south west)+(-.5,-.5) 
rectangle
([shift={(.5,.5)}]current bounding box.north east);
\end{scope}
\end{tikzpicture}
\end{document}