Du musst statt `text=$\alpha$` einfach nur `"$\alpha$"` schreiben:
schreiben (mit der `quotes`-Syntax):
\documentclass[headheight=19pt]{scrartcl}
%------------------------------------------------------------------------------
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{quotes,babel,angles}
%------------------------------------------------------------------------------
\tikzset{
winkel/.style={draw=gray,angle eccentricity=.6,angle radius=0.6cm},
mybox/.style={draw=gray,fill=white,align=left,text width=.9\linewidth}
}
%------------------------------------------------------------------------------
\usepackage{blindtext}% für Fülltext
\begin{document}
\begin{tikzpicture}[auto]
\draw
(4,0) coordinate[label=right:B] (B)
-- node{$c$}
(0,0) coordinate[label=left:A] (A)
-- node{$b$}
(1,2) coordinate[label=above right:C] (C)
-- node{$a$}
cycle
pic["$\alpha$", winkel]{angle=B--A--C}% Winkel bei Node A
%pic["$\cdot$", winkel]{angle=C--B--A}% Winkel bei Node B
pic["$\cdot$", winkel]{angle=A--C--B}% Winkel bei Node C
;
\end{tikzpicture}
\end{document}\end{document}
Oder, ohne die `quotes`-Syntax, muss es `pic text=$\alpha$` heißen:
\documentclass[headheight=19pt]{scrartcl}
%------------------------------------------------------------------------------
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{quotes,babel,angles}
%------------------------------------------------------------------------------
\tikzset{
winkel/.style={draw=gray,angle eccentricity=.6,angle radius=0.6cm},
mybox/.style={draw=gray,fill=white,align=left,text width=.9\linewidth}
}
%------------------------------------------------------------------------------
\usepackage{blindtext}% für Fülltext
\begin{document}
\begin{tikzpicture}[auto]
\draw
(4,0) coordinate[label=right:B] (B)
-- node{$c$}
(0,0) coordinate[label=left:A] (A)
-- node{$b$}
(1,2) coordinate[label=above right:C] (C)
-- node{$a$}
cycle
pic[pic text=$\alpha$, winkel]{angle=B--A--C}% Winkel bei Node A
%pic[pic text=$\cdot$, winkel]{angle=C--B--A}% Winkel bei Node B
pic[pic text=$\cdot$, winkel]{angle=A--C--B}% Winkel bei Node C
;
\end{tikzpicture}
\end{document}
