If I understand correctly, you want to save the local date / time at a specific question to a variable. Then you want to export that date / time variable later on at the analysis stage. Is that correct?
The below Perl Script should help ...
[%Begin Unverified Perl
In the pass-in fields, define "Q1TimeStamp" as a text data type. Change the date / time variable name to your preference.
You can place this Perl Script in the footer of the question you want the time stamp recorded at.
Note: You can set as many of these up as necessary.
And refer to this post if you want to extract certain elements of the date / time (e.g. the hour, the month, the date in a short format, etc.) ...