Here is a fairly easy way.

If your LaTeX document file does not already contain this line, add it in the top section before the begin{document} command:
\setpackage{fontspec}

Specify the main font for your score:
\setmainfont{Georgia} % or whatever font name you prefer. It must be installed already.

Then, also in the top section, call a macro once to define the style of the initial letter:
\grechangestyle{initial}{\setmainfont{Cloister Black BT}\fontsize{40}{40}\selectfont}

• grechangestyle is a macro for setting the styles of various elements. It's defined in the full manual for GregorioTeX, which you can find at
  https://github.com/gregorio-project/gregorio/releases/download/v6.0.0/GregorioRef.pdf
  (I trust you are running version 6.0.0 of gregorio.)
• initial specifies which element you're setting a style for.
• The code in the next set of braces, \setmainfont{Cloister Black BT}\fontsize{40}{40}\selectfont, selects the font you want, and specifies the size. This font must also be installed already.