According to http://jri-poland.org/psa/status.htm#AGAD, Lemberg/Lwow/Lviv births for 1913 do not appear in LDS (Mormon) microfilms, and there is ongoing fundraising for a project to create an index of birth, marriage, and death records for that city for a number of years including 1913 births.
Search for Lwow on that page to find more information, including the name and email address of the person coordinating the project.
It's also possible that you might be able to contact the archive which holds those records, either directly or through a researcher you hire, to search the records. Keep in mind that birth years listed in various documents (aside from original birth records) often aren't exact, as people might not have known their birth year or might have chosen to report a different year than they knew they were born. This certainly can apply to census records.
(Note that there may be alternate records which can help you find the information you're looking for. For instance, if you want to know the names of your grandmother's parents, the death record for her or one of her siblings might tell you those names.)