Referencing references within a quote I don't think this is quite the same as enter Referencing the reference? apologies if it is.

I am quoting someone in my thesis, they say:


  ... blah blah as noted in experiments by Tom (2013), Dick (2014) and Harry (2015)...


Should the three references in my quotation appear as in my bibliography even though they are not directly referenced by me but rather via the quote?
Chain referencing is not required unless you reference some information from the chain reference that is more than what is already referenced within the first citation. 

To add clarity, if you want to cite a matter from [A] which is cited as a quote from [B], then citing [B] would be appropriate. Otherwise if [A] adds some insights to [B], then citing just [A] would be fine. If in addition to this you derive some insights from [B] which is not noted in [A], then you must cite both [A] and [B].