Idea (from here): I use \symbol{idx}, where idx is the glyphs index.
To determine these two glyphs' indexes I had a look into ...texmf-var\luatex-cache\generic\fonts\otf\garamondpremrpro.lua and verified the result using fontforge as well as MS Word. All three report, that "34" and "39" should be the correct indexes - please see the following screenshots:



I also verified the general idea using the index of "blacksquare" (9632): Although the black square is set correctly, the quotedbl/quotesingle do not show up using the following code:
\documentclass{article}
\usepackage{fontspec}
\setmainfont{Garamond Premier Pro}
\newcommand{\blackSquare}{\symbol{9632}} %<-- works fine
\newcommand{\straightQuoteDbl}{\symbol{34}} %<-- does not work
\newcommand{\straightQuoteSingle}{\symbol{39}} %<-- does not work
\begin{document}
{\straightQuoteDbl}Hello World! \blackSquare\straightQuoteDbl
{\straightQuoteSingle}Hello World! \blackSquare\straightQuoteSingle
\end{document}
The code produces the following output:

Both quotes clearly differ from the results MS Word produces (see above).
System environment: Win 7 Pro 64-bit
TeX-Environment: texlive 2014 (frozen)
Font: Garamond Premier Pro Version 2.0.104
What am I doing wrong?

\setmainfont{Garamond Premier Pro}[RawFeature={-tlig}]or, to affect only parts of the document,\addfontfeatures{RawFeature={-tlig}}– Thérèse May 16 '15 at 16:39\newcommand{\straightQuoteSingle}{\addfontfeatures{RawFeature={-tlig}}\symbol{39}}. You should (re)write this solution into an answer. – May 16 '15 at 16:53