This works for me with XeLaTeX without any error:
\documentclass[symmetric,justified,marginals=raggedouter]{tufte-book}
\usepackage{lilyglyphs}
\usepackage{adjmulticol,tikz}
%alternative for the package lilyglyphs
\usepackage{fontspec}
\newfontfamily\emmentaler{emmentaler-11.otf}[Scale=MatchLowercase] %other fonts: emmentaler-11.otf emmentaler-13.otf emmentaler-14.otf emmentaler-16.otf emmentaler-18.otf emmentaler-20.otf emmentaler-23.otf emmentaler-26.otf
\newcommand\directFermata{{\emmentaler{}\symbol{"E132}}}
\begin{document}
\begin{figure*}[h]
\begin{tikzpicture}
\node[anchor=west,font=\tiny] (a) at (10.8cm,2.6cm) {\lilyGlyph{scripts.ufermata}};
\node[anchor=west,font=\tiny] (a) at (10.5cm,2.6cm) {\fermata\strut};
\node[anchor=west,font=\tiny] (a) at (10.0cm,2.6cm) {\directFermata\strut};
\end{tikzpicture}
\end{figure*}
\end{document}
But it does not work on pdfLaTeX since the command you use requires https://ctan.org/pkg/fontspec. This is also written in the second paragraph of the lilyglyphs' manual's section "1 Introduction" and in the first paragraph of the section "Quick Start".
You could also draw your own fermata with Tikz. It is one of the simpler music symbols. Or you could import it as pdf file, see https://tex.stackexchange.com/a/441624/128553. Both ways make you independent from https://ctan.org/pkg/fontspec in your main document.

! Undefined control sequence. l.9 ...est,font=\tiny](a)at(10.8cm,2.6cm){\lilyGlypg {fermata}\strut};. – Apr 29 '20 at 20:37\lilyGlyph-- note thehat the end. Presumably you also need to compile this with XeLaTeX. – Apr 29 '20 at 20:41\lilyGlyph, not\lilyGlypg. In general, if you ignore error messages in LaTeX you cannot expect to get a reasonable output. – Apr 29 '20 at 20:42