It follows 11 years of legal action between the FPA and DHSSPS.
Unlike the rest of the United Kingdom, abortion is only permitted in Northern Ireland under restricted circumstances, however the charity says there are no good practice guidelines to enable clinicians and women to interpret the law, and thus it is not clear when it can take place.
"The current situation is unacceptable," said Dr Audrey Simpson OBE.
"Neither health professionals or women know when an abortion is permissible and this has devastating effects on women's health and well-being and the standard of care professionals are able to give them.
"We have spent over a decade on this issue and we will not give up until the Northern Ireland Assembly provides the guidance that is so urgently needed."
Guidelines for doctors were published in 2009 but were revoked after a legal review.
The DHSSPS said it would only issue guidance compatible with the law.
Abortion remains a controversial topic in NI, with the region's first private sexual health clinic opening last October amid protests by pro-life campaigners.
They say there is no desire for the services offered at the Marie Stopes clinic and argue that the law on abortion is perfectly clear and requires no further guidelines.
The hearing at the High Court in Belfast is expected to finish on Thursday.