Henri Menke provided a partial solution:

\defineinitial[two][method={first,auto}]
\setuphead[chapter][
  after={\placeinitial[two]},
  page=no,
]

\starttext
\chapter{Harriet Tubman}
"Every great dream begins with a dreamer. Always remember, you have within
you the strength, the patience, and the passion to reach for the stars to
change the world.
\chapter{Eleanor Roosevelt}
Great minds discuss ideas; average minds discuss events; small minds
discuss people.
\stoptext

However, the leading quotation mark is gobbled. Is there any generic way of detecting a leading quotation mark while keeping initial caps on the first character?

Thank you!