Can I quote a full paragraph which contains some other citations?
I assume that the citations in this paragraph are not material to your dissertation.  I also assume that they are also not material to the quoted paragraph.  If so, then you can omit the citations and note that you have done so by adding "[citations omitted]" immediately after the quoted paragraph.

For example, if this is the original quote:


  "Counterfactuals are thus temporally asymmetric: Events in the counterfactual world prior to the antecedent would be more similar than would subsequent events to those of the actual world (e.g., Edgington, 2004; Lewis, 1979). This asymmetry may be related to the finding that people have an easier time imagining counterfactual alternatives to later events in a sequence than to earlier events (Miller &amp; Gunasegaram, 1990). The earlier events seem fixed, the later events more mutable, at least when no further relations constrain them (see Byrne, 2005; Kahneman &amp; Miller, 1986; Parker &amp; Tetlock, 2006)."


You could quote it this way:


  Counterfactuals are thus temporally asymmetric: Events in the counterfactual world prior to the antecedent would be more similar than would subsequent events to those of the actual world. This asymmetry may be related to the finding that people have an easier time imagining counterfactual alternatives to later events in a sequence than to earlier events. The earlier events seem fixed, the later events more mutable, at least when no further relations constrain them." [citations omitted]


EDIT
By "material to the dissertation or quoted paragraph", I mean cases where the citations are important or central, and leaving them out would leave a void for the reader.  For example, let's say your dissertation chapter was a comparison of two rival theories, and your quotation made specific points about one or two of the prime authors of those theories.  It would then be material and including the citations in the quote would be valuable to the reader.  Another case: let's say your quotation was a commentary on what one or more authors said or did.  Leaving out the citations would make it ambiguous for the reader as to what source material your quotation referred to.