I'm neither TG myself (I came here to visit to gather information for a TG character) nor a lawyer (so don't treat this as legal advice), but do you live in an area where TGs are protected by anti-discrimination laws in terms of employment? If so, them asking for previous names at the initial application stage may be a bit dicey for them. I'd compare it to them asking (anyone whether TG or not) for their birthdate at the same application stage (asking if you are at least a certain minimum age in order to comply with child labor laws is acceptable though). What I mean of "application stage" for both the name and birthdate is if the information is asked later for background check purposes and will not be used to influence hiring decisions, then give it then. Another instance where you may need to out yourself via the name is if you have references that don't know about your transition or educational records that you need to give which you weren't able to change.
If you don't live in an area covered by anti-discrimination laws protecting TGs in employment, then them asking for the previous name is probably perfectly legal. If you don't live in the U.S. this probably won't apply to you (e.g. in some countries asking for birthdate is legal).
ETA: I am not suggesting whether or not the OP should mention the information, but rather that in areas where transgenders are a "protected class" this question* might be considered "taboo" (like the aforementioned question about birthdate). *If worded in a way that does not out a TG (e.g. asking for only maiden or other last names used, or specifically mentioning that TGs do not need to reveal pre-transition names), this statement would not apply. I do not know of any court rulings one way or the other on this issue.