Since we have $\mathscr{F}$, the script version of an English alphabet, do we also have the script version of a Greek alphabet such as $\Omega$?
I have tried $\mathscr{\Omega}$ but nothing had happened.
Since we have $\mathscr{F}$, the script version of an English alphabet, do we also have the script version of a Greek alphabet such as $\Omega$?
I have tried $\mathscr{\Omega}$ but nothing had happened.
The question is a bit vague. If you find some font which has such symbols, you might use them with use of the package fontspec.
But to answer the question "do we have script Greek symbol?" in the meaning of predefined in LaTeX or some package I say: no.
The package unicode-math handles 6 very common Math fonts you will find in many of the examples here and here is their list of normal weight alphabetical symbols from their documentation:

No Greek script can be found here. As there is no Unicode-block for such symbols either, I guess it will get difficult to find something.
My recommendation: Search for some font of hand-written Greek, download and install it, and do:
% arara: lualatex
\documentclass{article}
\usepackage{fontspec}
\usepackage{mathtools}
\newcommand*{\mscrOmega}{\text{\fontspec{yourFont.otf}\symbol{"03A9}}}
\begin{document}
$\mscrOmega$ \mscrOmega
\end{document}
To complement LaRiFaRi's answer, Microsoft's Gabriola font do have some capital Greek letters that look handwritten, but they just miss a slight italic aspect. A clean way to resolve this would be to use XeLatex with Gabriel font and fake the slant with TikZ. If you just need a single letter, you could also fake the slant with some design/vector software (say inkscape) and import the resulting symbol. Result with a capital cursive theta: